Who is Britain's favourite American dramatist? One year it seems to be Arthur Miller, the next it's David Mamet. Right now, Tennessee Williams is having a moment. Rachel Weisz opens in A Streetcar Named Desire tonight, at the Donmar in London. In December, a Broadway African-American Cat On a Hot Tin Roof, starring James Earl Jones and Adrian Lester, comes to the West End. And, in between, there is the European premiere of a forgotten 1937 play, Spring Storm, at the Royal & Derngate in Northampton. But, for all our enthusiasm for Williams, I think we still get him subtly wrong. He is most often dubbed a "psychological" dramatist, but this ignores his social and political radicalism – as well as his rich talent for comedy.

Of course, perceptions of Williams have evolved over the years. When Streetcar was first seen in London in 1949, in a production directed by Laurence Olivier and starring Vivien Leigh, Williams was viewed as a kind of filthy American sleaze-merchant. The confrontation of Blanche Dubois and Stanley Kowalski sent the British press into a tizzy: Logan Gourlay in the Sunday Express spoke for many when he condemned the play as "the progress of a prostitute, the flight of a nymphomaniac, the ravings of a sexual neurotic". The play was attacked in Parliament as "low and repugnant", and by the Public Morality Council as "salacious and pornographic". When Cat On a Hot Tin Roof had its British premiere in 1958, it had to be presented under the polite fiction of a "club performance" – lest the broader public be corrupted by the discreet suggestion that its hero, Brick, is gay.

Thankfully, we have got over the idea that Williams is a sexual sensationalist. Now, we tend to emphasise his lyric, compassionate side. He has been described as "the poet of lost souls", and we tend to dwell on his empathy with the spiritually wounded. In this reading, the keynote line is that of the itinerant Val in Williams's Orpheus Descending, who says: "We're all of us sentenced to solitary confinement inside our own skins, for life!"

I'd be the last to deny Williams's poignant poetry and echoing loneliness, but we overlook the more robust side of his talent. His plays offer a devastating portrait of the ugly prejudices of the American south; they acknowledge economic realities; and, in the spirit of his admired Chekhov, are as comic as they are tragic. For evidence of Williams's strong social conscience – the product of his Episcopalian upbringing – you have to look at the life and letters, as well as the work. Travelling through the American south-west in 1939, he listened attentively to the stories of impoverished, workless families, and shared his meagre rations with their children. He also loathed racism. Appalled that he was unable to prevent The Glass Menagerie playing to all-white audiences in the capital, in 1947 Williams wrote to the New York Times that "any future contract I make will contain a clause to keep the show out of Washington while this undemocratic practice continues". He was influenced by the theories of Brecht's collaborator Erwin Piscator, a German emigre who ran the Drama Workshop in New York and was instrumental in staging an early Williams one-acter, The Long Goodbye, in 1940. Williams's early plays, in particular, often reveal a strong social purpose – a point proved by Trevor Nunn's National theatre revival of Not About Nightingales. Written in 1938, this play vividly dramatised a group of prisoners challenging the Mussolini-like power of a brutal warden. This anti-fascist instinct was one of Williams's guiding principles.

In this sense, A Streetcar Named Desire is a test case. For some, the play is a vindication of Stanley Kowalski, who finally gets his revenge on his sister-in-law, Blanche, who has intruded on his happy New Orleans home with her fancy airs (no surprise to find John Osborne among the fervent Stanleyites, writing that "the female must come toppling down to where she should be – on her back"). But there is another way of looking at the play, brilliantly articulated by the American director-critic Harold Clurman in 1948. For Clurman, Blanche represents "the martyred poetic instinct and aristocracy of feeling", while Stanley is all muscle, lumpish sensuality and crude energy: his "mentality provides the soil for fascism, viewed not as a political movement but as a state of being". I think the play is not quite as black-and-white as that, and requires a more delicate balance of sympathy; but he was spot-on in seeing Streetcar as "wholly a product of our life today". It was a direct response to late-1940s America, in which a macho materialism prevailed over spiritual and artistic values.

Before Williams entered what he himself called his "stoned age" of the 1960s, when drink and drugs stifled his talent, his subject was America in the biggest, most political sense. Even Camino Real, first performed in 1953 and apparently showing Williams at his most woozily symbolic, is a vivid metaphor for 1950s America. Set in a crumbling plaza, the play brings together a group of doomed romantic idealists, including Don Quixote, Lord Byron, Casanova and Marguerite Gautier. Far from being a hazy phantasmagoria, the play was a direct and passionate response to an America where, as Williams said, "the spring of humanity had gone dry"; the fascist demagoguery of Senator McCarthy was all-pervasive. Intriguingly, Camino Real appeared in the same year as Arthur Miller's The Crucible, and was just as opposed to the ethos of the times; Williams was among the first to protest at the withdrawal of Miller's passport by the US state department. Williams remained an implicitly political writer up to and including Orpheus Descending, first performed in 1957. Here, the heroine discovers that her father was killed by the Ku Klux Klan for his failure to practise racial discrimination. Williams is known as an explorer of interior landscapes; what we miss is his capacity to record, and to fight against, any form of prejudice, intolerance or oppression.

At his best, Williams pinned down America at its worst. But for all his empathy, he saw the human condition – and his own especially – as faintly absurd. I have some evidence for this, in my only encounter with him, on Radio 4's Start the Week in 1978. I had been asked to interview Williams about his nakedly autobiographical play, Vieux Carré, which had just opened in London. I had been briefed to begin with a summary of the plot, and as I described the struggles of the play's writer-hero, living in a New Orleans apartment-block surrounded by starving gentlefolk, alcoholics and deadbeats, I heard a steadily growing rumble of laughter from across the table. At first, I thought it was a comment on my inept synopsis. Gradually, I realised it was the sound of a writer who saw his own remembered tribulations as hilarious. I recalled that a year earlier he had been asked by the management to leave a performance of The Glass Menagerie at the Shaw theatre in London, because his incessant hilarity at this memory of his own youth was disturbing the rest of the audience.

Williams's gift for comedy is partly temperamental, and partly a product of a southern inheritance that extends from Mark Twain to Carson McCullers. It pervades much of his work. You can see Cat On a Hot Tin Roof as a tragedy about self-delusion, in which the hero, Brick, can no more admit to his homosexuality than his father, Big Daddy, can to his cancer. But, as Howard Davies's superb production at the National in 1988 reminded us, the play is also very funny. It opens with a hilarious monologue by Maggie the Cat, in which she protests against the "no-neck monsters" bred by her endlessly fecund sister-in-law. Williams could also be ribald and Dionysiac. In The Rose Tattoo, a devout American-Sicilian widow, Serafina, is aroused from a three-year period of mourning by the arrival of a sexy, muscular buffoon. The sight of Zoë Wanamaker, in the most recent revival, struggling to get out of her panty-girdle in time to keep an appointment with her lover was hilarious. At a deeper level, of course, the play is about the force for renewal in human life and the need to break with the past.

Some of Williams's later work borders on self-parody, but the best of his plays will endure the ravages of time, for a number of reasons: there is his capacity to write great parts for actors, for one thing; his instinctive love of the marginalised and defeated; and his fervent opposition to any form of tyranny, whether domestic or political. As Williams once wrote (significantly, he used capital letters): "to be an Artist is to be a Revolutionary", whose words are always feared and misunderstood. It's time to acknowledge that, inside the complex state of Tennessee, existed a natural rebel.

'As flamboyant as an orchid' – Williams on his leading ladies

On Vivien Leigh

Leigh is not only the officially appointed first lady of the London theatre, but several other things of equal or greater importance: an actress of great talent which has steadily grown through meeting the challenge of many classic roles, while still appearing so masterfully in such American films as Gone With the Wind and Streetcar. At present she is appearing in a film based on my novel The Roman Spring of Mrs Stone, a part she accepted despite its being an ageing actress, infatuated with a young adventurer.

Vivien's beauty is as delicately flamboyant as an orchid. When she takes the stage, she commands it as if she first arrived there suspended from the bill of a stork. Vivien is incomparably graceful, she moves like a marvellous dancer, on or off stage, and has an instinct for doing and saying the right thing to put you at ease even when you know you are making a fool of yourself. All of these wonderful gifts she has given with no apparent regard for her personal vulnerability: in other words she is not only a stunning actress but a lady with the most important part of that intricate composition, which is kindness of heart.

On Elizabeth Taylor

Taylor is one of the great phenomena and symptoms of our time in America. Everything she has done has been startling and sometimes implausible. It's hard to guess what Liz wants out of life: the million-dollar contracts and diamond necklaces or the exercise of what is probably the finest raw talent on the Hollywood screen. She has a deeply moving response to fellow artists in trouble, and so I'm inclined to believe she is more interested in the creative work than the loot.

She thoroughly understands the system she is caught in, and she is not to be bossed or intimidated by it. Hollywood moguls have met their match. I've heard her phone conversations with them: mutual understanding and respect. Naturally I hope that this girl, this cross between a flower and the rock it sprang from, will discover that her greater satisfaction will come from the disciplining of her talent in the stricter conditions of our theatre.

As for Liz's performance in Suddenly, Last Summer, it demonstrated her ability to rise above miscasting. She was well cast as Maggie in Cat on a Hot Tin Roof, and should have got her Oscar. But it stretched my credulity to believe Liz wouldn't know [in Suddenly, Last Summer] that she was "being used for something evil". Liz would have dragged Sebastian home by his ears, and so saved them both from considerable embarrassment.

Extracted from Five Fiery Ladies, 1961, now collected in New Selected Essays: Where I Live (New Directions)