With 4 July in the offing, it seems appropriate to take a look at three pivotal American independent film-makers and examine how they have sought to break with the Hollywood mainstream in the six decades since John Cassavetes lost interest in simply acting in movies.

The godfather of indie cinema is something of an acquired taste, as his largely improvised pictures can occasionally lose focus and drift into self-indulgence. But no other American director challenged the conventions of the Hollywood narrative with such fearless experimentation. As Cassavetes said himself: ‘I like to act in films, I like to shoot them, I like to direct them, I like to be around them… It doesn't make any difference whether it's a crappy film or a good film. Anyone who can make a film, I already love. But I feel sorry if they don't put any thought in it because then they missed the boat…you must have the courage to be bad - to be willing to risk everything to really express it all.'

Born in New York in 1929, Cassavetes trained at AADA and taught at the Actors Studio while taking paying jobs on the big and small screen. At the age of 28, he used his fee from the TV series Johnny Staccato to fund his directorial debut, Shadows (1960), a racial drama shot in 16mm on New York streets for a mere $40,000. Combining techniques gleaned from the nouvelle vague with a script that was derived from much improvisational experiment, the story of light-skinned Leila Goldoni's romance with white bohemian Manhattanite Anthony Ray is much more briskly paced than Cassavetes's later outings. However, the subject matter is every bit as contentious and this simmering monochrome melodrama scored to cool jazz by the great Charles Mingus won the critics prize at the Venice Film Festival and proved pivotal to the emergence of New American Cinema in the 1960s.

The music was even more to the fore in Too Late Blues (1961), which was originally written for Montgomery Clift, but ended up starring crooner Bobby Darin when the mercurial Method actor withdrew at the eleventh hour. His presence is much missed, as Darin lacks the intensity to convince as the leader of a struggling combo gigging its way around Los Angeles in search of its sound and its audience. Darin is persuaded to lose faith in agent Everett Chambers by aspiring chanteuse Stella Stevens. But, even though Stevens helps the band land a recording contract, Darin is dismayed to discover that she works as a prostitute to pay the rent. She is equally disillusioned by his inability to stand up to pool-hall thug Vince Edwards and they drift apart, as Darin falls in with wealthy countess Marilyn Clark and starts playing muzak in soulless uptown clubs.

With Benny Carter, Shelly Manne and Jimmy Rowles among those playing on the 17 new numbers included on the soundtrack, this should have found its niche among jazz aficionados. But the script concocted with Richard Carr was insufficiently down and dirty and Cassavetes only managed one more directorial assignment (1963's A Child Is Waiting, which teamed Burt Lancaster and Judy Garland in a mawkish Stanley Kramer melodrama set in a school for developmentally challenged children) before he demonstrated a new maturity with Faces (1968), which earned an Oscar nomination for its screenplay, as well as Best Supporting nods for Seymour Cassell and Lynn Carlin, who look on as Gena Rowlands's marriage to John Marley begins to fall apart. While many applauded this intense approach to resolutely adult subject and Cassavetes's fascination with the art of performance, others complained about the camera being subservient to the actors and the director's tendency to dissipate the dramatic impact of scenes by allowing them to overrun in the hope of capturing fresh inspiration.

By 1974, Cassavetes had not only found his style, but also an ensemble capable of conveying his vision. A Woman Under the Influence (1974) may not be perfect - it meanders over 155 minutes and Gena Rowlands periodically descends into soap operatics - but it exemplifies the way in which Cassavetes gave his actors room to create their characters and conflicts, while also showcasing the restless camera technique that maintained the action's energy and immediacy. Rowlands is ably matched by Peter Falk, as the Italian construction worker who becomes increasingly bemused by his wife's erratic behaviour in front of his workmates and kids. Indeed, his controlled performance contrasts bristlingly with Rowlands's blowsier display. But she also manages moments of restrained poignancy, particularly after she returns to her home in the Los Angeles suburbs after spending six months in an institution and finds it almost impossible to pick up the pieces of her permanently altered life.

Cassavetes was clearly in awe of his wife's talent and he devised Opening Night (1978) specially for her. Although its story about a career-obsessed actress bears traces of All About Eve and Sweet Bird of Youth, this intense drama could easily be entitled A Woman Under the Influence II. Rowlands landed an Oscar nomination for her study of an ageing woman crumbling as her life slips away from her and she won the Best Actress prize at Berlin for this courageous display, as the star whose role in Joan Blondell's new play reflects her own concerns with losing all she has worked for. Whether succumbing to drunken self-pity or being tormented by the memory of the death of a young fan, Rowlands sustains an aura of desperation that is often excruciating to watch. Cassavetes nails the backstage milieu. But the detached sentience of the direction prompts a feeling of hollowness that serves only to  emphasise the overall air of calculation.

A similar problem dogs The Killing of a Chinese Bookie (1976), which is as much a parable about Cassavetes's struggles to raise the finance to make his films as it is about LA strip club owner Ben Gazzara agreeing to commit murder to pay back the $20,000 he owes to mobster Seymour Cassell. At its best, film noir moves at a fair clip, with dialogue being bitten off and spat out by hard-boiled characters whose every word and gesture represents an effort to rouse themselves from their in-grained cynicism, world-weariness or simmering malevolence. Cassell's scheming crook could have sidled out of the pages of Chandler or Hammett. But Gazzara very much remains a resident of Cassavetes's rarified world, even after he discovers that his target is not a Chinese bookie, but a West Coast gang boss surrounded by knuckle-dragging bodyguards. Cassavetes obviously intended to replace B-movie briskness with a genre-subverting languor that would heighten the suspense of Gazzara's plight. But the naturalism achieved by dwelling on the pathetic neediness of the lowlifes patronising the gambling joint is undermined by the artifice that inexorably dominates proceedings. Cassavetes evidently realised that the picture misfired, as he re-edited it from 135 to 109 minutes in 1978. But it remains as flawed as it's fascinating.

Having struggled to come to terms with his Mormon father's suicide when he was just 12 years old, Utah-born Hal Ashby found it difficult to sustain relationships and was married five times. But he made few foes in an industry renowned for its back-stabbing and the fact that he could coax memorable performances from such unpredictable characters as Jack Nicholson, Jane Fonda and David Carradine speaks volumes for the geniality of his temperament and his grasp of screen craft. Indeed, all lazy references to Ashby as a trippy hippy who was too stoned to function on the set and then edited himself into knots during post-production should be banished forever.

Moreover, there should be no further excuse for denying Ashby his place among the most important American film-makers of the post-studio era. He was an exceptional editor, winning an Oscar for his work on Norman Jewison's In the Heat of the Night (1967) and devising the split-screen technique that made The Thomas Crown Affair (1968) so chic. Yet Ashby was also an inveterate maverick and, in Being There, he couldn't resist parodying the famously sensual chess match between Steve McQueen and Faye Dunaway, as Peter Sellers and Shirley Maclaine make their moves on each other while watching the picture on television.

This sense of mischief also informs Ashby's directorial debut, The Landlord (1970), African-American actor Bill Gunn's adaptation of a Kristin Hunter novel that stars Beau Bridges as the scion of Long Island bigots Walter Brooke and Lee Grant, who purchases a tenement in Brooklyn's Park Slope district with the intention of evicting the squatting occupants and creating himself a trendy bachelor pad. However, much to his surprise, he finds himself becoming fond of fortune-teller Pearl Bailey and embarks upon affairs with both art student-cum-dancer Marki Bey and Diana Sands, who is married to the pugnacious Lou Gossett, Jr. Indeed, Bridges becomes so ashamed of his folks and his past prejudices that he even starts sympathising with black segregationist, Melvin Stewart. But Bridges's standing in the community changes when he gets Sands pregnant and Bey disowns him.

Despite its sharp social satire and moments of touching poignancy, this fascinating snapshot of post-Civil Rights America suffers from the odd longueur and the odd moment of sitcom cornball. However, Bridges exudes confused charm, as the spoilt brat confronting real life for the first time, while Sands and Bey ably convey the frustrations of falling victim to both black and white chauvinism. The acting honours, however, go to Bailey and Grant, whose sharp tongues anticipate the role that Ashby would create for the veteran Ruth Gordon in Harold and Maude (1971).

Twentysomething Bud Cort lives with wealthy widowed mother Vivian Pickles, who abhors his habit of feigning suicide, driving around in a hearse and attending the funerals of complete strangers. She hopes that marrying him off will help him mature, but Cort is only interested in 79 year-old Ruth Gordon, whose zest for life contrasts sharply with his morbidity. In addition to showing him how to play the banjo, Gordon also teaches Cort about music and art and convinces him to seize each and every day. But she says little about her life as a young bride in pre-war Viena or the concentration camp tattoo on her arm, as she falls in love with Cort and conspires to ruin the blind dates that Pickles has arranged through an agency.

Judy Engles is frightened away with a simulated immolation, while Shari Summers flees thinking Cort has severed a hand. But actress Ellen Geer sees his fake suicide as an excuse to trot out a little Romeo and Juliet before plunging the stage dagger into her own breast. However, genuine tragedy looms, as Gordon has always said that 80 was the perfect age to die and she bows out at the end of a surprise birthday party, in spite of Cort's frantic efforts to save her. But she has taught him well and he pulls one last stunt before heading off into the sunset at the end of Colin Higgins's extraordinary screenplay, which started out as his course thesis at UCLA.

It's hard to imagine why this failed to connect with audiences in the early 70s, as it is now regarded as a cult classic. The byplay between Cort and Gordon is both unnerving and exhilarating and how they missed out on Oscar nominations is anybody's guess. Pickles also excels, as she stumbles across her son's latest mock hanging, drowning or shooting. But the key to the picture is Ashby's ability to make two such mismatched ghouls seem adorable together, although John A. Alonzo's superb cinematography and the Cat Stevens songs on the soundtrack play their part, too. There are sluggish moments and a whiff of sentimentality creeps in towards the end to complement the hint of chauvinism that pervades the date sequences. But bad taste has rarely seemed so charming.

By contrast, Hal Hartley is renowned for the wry wit of his sophisticated dialogue and a deadpan timbre that often invites comparison with fellow indies Whit Stillman and Joel and Ethan Coen  Raised in a working-class Long Island neighbourhood, Hartley went into advertising after leaving film school and produced a number of shorts before making his feature bow with The Unbelievable Truth (1988), a pitch black comedy that was filmed on a shoestring budget in 12 days and introduced several members of the stock company who would become familiar faces in Halworld over the next few years. But what most impresses here is the teasing control Hartley retains over his script, as the black-clad Robert John Burke arrives in upstate New York amidst rumours that he is either a mass murderer or a priest.

Disillusioned student Adrienne Shelly persuades father Chris Cooke to hire Burke as a mechanic at his garage, in the hope that the favour will stop her obsessing about the apocalypse. But Shelly sees Burke as an ally in her battle to avoid going to Harvard and also uses him to upset recently dumped boyfriend Gary Sauer. However, the rumour keeps circulating that Burke killed the father and sister of burger bar waitress Julia McNeal and it is only after Shelly falls in with sleazy underwear photographer David Healy that the `unbelievable truth' finally emerges about the night that McNeal's sister died in a car crash and her father supposedly fell down the stairs.

With its droll plot and deceptively perceptive asides on family life, the arms race and free will, this anti-love story won a prize at the Sundance Film Festival and announced Hartley as a talent to watch. He reinforced his reputation with the surprisingly similar Trust (1990), in which Shelly's high-school student alienates mother Merrit Nelson and sister Edie Falco by causing her father to have a fatal heart attack when she breaks the news she is pregnant. Given that she has also been dumped by quarterback boyfriend Gary Sauer, Shelly is ripe to fall into step with Martin Donovan, an electronics repairman who is afraid of tinkering with television sets and carries around a grenade in case he feels like committing suicide.

The smiles come when Shelly introduces the twitchy Donovan to her family and she takes exception to the bullying ignorance of his blue-collar father, John Alexander MacKay. But, while some critics insisted that this odd couple saga was a lower-rung variation on Woody Allen's studies of domestic dysfunction and nerdy neurosis, it is only fitfully amusing. Indeed, the humour occasionally feels forced, especially when it involves a notorious baby snatcher and a rapacious convenience store clerk. But the performances are splendidly off-kilter, with Karen Silas cameoing as a no-nonsense nurse and Falco stealing scenes as the embittered sibling who has just lost custody of her children to her conniving husband.

Two years later, Hartley returned with Simple Men (1992), which has a distinct feel of Aki Kaurismäki about its story of Long Island brothers Robert John Burke and William Sage, who join forces to find their long-lost father, John Alexander MacKay. Having recently been cheated by his girlfriend, computer thief Burke has vowed to wreak his revenge on the next blonde he meets. But Sage is keen to track down MacKay, who has been on the run for 23 years since he abandoned his career as a baseball shortstop with the Brooklyn Dodgers to become an anarchist bomber who killed seven innocent people in a plot to blow up the Pentagon in 1968. The FBI are also on his tail, but the pursuit is complicated by a meeting with epileptic Romanian Elina Löwensohn and her friend Karen Silas, who runs a roadside café. Despite developing a crush, Sage is certain that Löwensohn knows where MacKay is hiding. But Burke is more interested in seducing Silas from under the noses of her bruising ex-husband, Joe Stevens (who has just been released from the slammer), and his best buddy, Martin Donovan, who owns a fishing boat.

As if the twisting tale isn't enough, Hartley revels in confronting the principals with such quirky characters as the security guard who gives Burke a statue of the Virgin Mary to keep him safe (Richard Reyes), the petrol jockey who practices French on his customers (Mark Chandley Bailey) and the sheriff with a penchant for emotive speeches (Damian Young). But, for all the idiosyncratic intelligence of the screenplay and the laconic lunacy of the performances, the flippant philosophising sometimes smacks of smugness, as does the homage to the impromptu dance sequence in Jean-Luc Godard's Bande à part (1964).

The French connection continued into Amateur (1994), a `metaphysical thriller' that starred Isabelle Huppert as a lapsed nun who has convinced herself that she is a nymphomaniac with a talent for writing smutty fiction. She meets amnesiac Martin Donovan in a café after he wakes up bemused in an alleyway. But, as she nurses Donovan back to health in her apartment, his memory slowly starts to return and Huppert realises she is harbouring a vicious pornographer who has survived being pushed out of a window by his porn star wife, Elina Löwensohn. However, Löwensohn is certain that she has killed Donovan and uses his identity to try and pull a fast one on a mysterious arms dealer from Amsterdam, who has dispatched accountant Damian Young and philosophy-spouting palookas Chuck Montgomery and David Simonds to dispose of Donovan.

Once again, it seems easy to detect comparisons between this offbeat romp and the mischievous pictures of Aki Kaurismäki. The inimitable Huppert is touchingly bemused as the innocent pitched into a milieu she scarcely understands, while Donovan slips creditably between mild-mannered and menacing. But, while he enjoys himself parodying the conventions, clichés and caricatures of film noir and makes some salient points about identity and morality and exploitation and redemption, Hartley struggles to engage the audience with the macguffin of a floppy disk that contains evidence of high-level corruption that could bring down governments across the Atlantic.

As ever, the dialogue is polished, but the mock gravitas is often overdone and Hartley would similarly miss his step with Flirt (1995), a triptych that centred on romantic entanglements in New York, Berlin and Tokyo. However, he returned to form with Henry Fool (1997), which turned out to be his biggest commercial success and one of his most favourably reviewed outings after it took the prize for Best Screenplay at Cannes.

Nobody knows quite what to expect when writer Thomas Jay Ryan arrives on the doorstep and persuades Maria Porter to let him lodge with her family while he works on a magnum opus entitled Confusion. Within days of his arrival, however, Ryan has slept with both Porter and her daughter, Parker Posey. But he has the most impact on Posey's brother, James Urbaniak, a socially inept binman who starts writing in a journal and amazes Ryan by producing screeds of misspelt naive poetry in otherwise impeccably composed iambic pentameters. Ryan persuades Urbaniak to post his verses online and he quickly attracts a passionate following. Yet, as he is being declared a genius, Posey informs Ryan that she is pregnant and he agrees to marry her and take over Urbaniak's job on the bins to support the family.

Amusingly inverting the basic premise of Pier Paolo Pasolini's biting satire, Theorem (1968), this is one of the first films to question whether the Internet has been responsible for dumbing down rather than democratising the worlds of arts and letters. Debuting in the title role, stage actor Thomas Jay Ryan exemplifies the kind of easy intelligence that has been consigned to the margins by the opinionated piffle purveyed by prolific hacks who would never have been published in a month of print-dominated Sundays. In many ways, Ryan deserves to fall from grace for his preening, patronising and pontificating. But, while few would begrudge Urbaniak his rise to literary fame (or his Nobel Prize), especially after he was dismissed so callously as a peddler of filth by one editor, one can't quite condemn Ryan out of hand.

Some critics lambasted Hartley for lurching between moments of pretentious navel gazing and crude gross-out slapstick. Yet, while Ryan's slide into ignominy is both drawn out and more than a little misogynist, this is a shrewd study of basic human nature and an unsuspecting society on the cusp of a communication revolution. Sadly, Hartley failed to live up to expectations with either The Book of Life (1998) or No Such Thing (2001) and it wasn't until he returned to the perils of technology and the ease with which people and the media can be manipulated by unscrupulous, corrupt and greedy tycoons that he almost got back on track with The Girl From Monday (2005).

Set in a near future in which the United States is run by the Major Multimedia Monopoly that ousted the government in the Great Revolution and turned the country into a giant business, the action centres on Bill Sage, an executive in the Brutal Youth Children's Wear division of Triple M who is regretting the fact that he helped draft the Human Value Reform Act that connects sexual performance to credit rating. The company has already commoditised every aspect of life and given all citizens their own bar code tattoo in order to monitor their daily consumption of everything from snacks to aphrodisiacs. But, by declaring sex for love or pleasure to be a crime of perversion, Sage believes ambitious boss DJ Mendel has gone too far and he tries to use his position as secret head of the anti-capitalist Partisans to bring down the Dictatorship of the Consumer.

However, Sage's own credit rating has taken a dip after he was spurned by Sabrina Lloyd, who is much more attracted to teenage Partisan, Leo Fitzpatrick. But her passion proves her undoing and she is sentenced to two years' hard labour as a high-school teacher. While mooching around the premises, she chances upon Henry David Thoreau's back-to-basics novel, Walden, and decides to join the rebels. Sage is equally keen to undermine Triple M, but his focus is deflected by the arrival of Tatiana Abracos, an alien from the planet Monday who is aware that Sage is also an extraterrestrial from the Star 147X constellation. She wants him to abandon Earth, but he is less keen to resume his existence as a bodiless part of a communal consciousness and Abracos's efforts to lure him away lead her to have her own first experience of human emotion.

Hartley described this inventive, but insubstantial offering as `a fake science fiction film about the way we live now'. It's certainly packed with shrewd borrowings from Thoreau, George Orwell and Aldous Huxley. But for all the acuity of Inbal Weinberg's chic production design, Sarah Cawley-Cabiya's canted imagery and Steve Hamilton's slick editing, this is a high-concept arthouse muddle that is far too pleased with its satirical set-up to bother with narrative coherence or credible characterisation. Throwaway gags about counter-revolutionaries being sent to run concession stands at theme parks on the Moon are all very well. But Hartley misses the opportunity to warn audiences about the potential dangers of surrendering information and individuality to unregulated behemoths like Google, Amazon and Facebook and, consequently, this lacks the potency of Jean-Luc Godard's acute exercise in futurist parody, Alphaville (1965).