The conductor Serge Koussevitsky described seeing Gershwin in full flow at the keyboard. 'As I watched him, I caught myself thinking, in a dream state, that this was a delusion; the enchantment of this extraordinary being was too great to be real.' Gershwin may never have learnt to read music fluently, but that didn't matter. His keyboard wizardry was an alchemy of finger dexterity, memory and a phenomenal capacity for instant invention.

He was a compulsive performer. At parties, he would commandeer the piano and play variations on his show-tunes into the night. When even his mother told him not to overdo it, he just said, 'If I don't play at a party I don't have a good time!' His listeners might be society celebrities and starry-eyed flappers, draped around his piano in Manhattan during the Prohibition years of the Twenties. Equally, they could be Ravel and Prokofiev in Paris or Alban Berg in Vienna. Open-minded towards the newest trends, these composers made a point of asking Gershwin to play for them – and duly experienced the 'Koussevitsky effect'.

Having left school at 15, Gershwin became the youngest 'piano pounder' (someone who has to publicise the publisher's tunes by performing them) on Tin Pan Alley. But he never performed the sheet music as printed. This gave his playing extra 'zip' and provided an arsenal of pianistic tricks to use in his own pieces. He abandoned his job because 'the popular song racket began to get definitely on my nerves.' On the way to achieving part of his ambition – to write hit musicals - he won fame and fortune through his one true 'pop' song, 'Swanee', popularised by Al Jolson.

Established as a major figure on Broadway by his early twenties, Gershwin became a truly 'hands-on' pianist and composer. At rehearsals he would effortlessly change key to suit his vocalists, tailor his material at the whim of a producer, or craftily rescue a song from a failed musical and re-invent it for another. He 'played' with notes in the truest sense – he had fun with his music, improvising witty variations on his show-tunes by the hour. The songwriter Burton Lane remarked: 'He was one of the few composers who had a real sense of humour.'

He had listened to ragtime pianists in Harlem as a boy, assimilating their styles and mannerisms along with the blues-laden soulfulness and rhythmic ingenuity of Negro jazz. He wanted to extend the potential of jazz into larger forms, but he was not the first to use it in concert music. Milhaud, Satie and Stravinsky were among the European composers who were attracted by ragtime and American dance-crazes such as the foxtrot. They had flirted with jazz before 1924, when Rhapsody in Blue climaxed an experimental jazz concert by bandleader Paul Whiteman. Whiteman's publicity machine dubbed him 'The King of Jazz', but neither he nor Gershwin were close to the black folk-roots of jazz. Whiteman's was an all-white band; his carefully rehearsed, sometimes brash arrangements were intended for dancing and they allowed little room for the improvised spontaneity of real jazz.

In a 1933 article called 'The Relation of Jazz to American Music', Gershwin defined his fundamental creed behind his concert works: 'Jazz I regard as an American folk music; not the only one, but a very powerful one which is probably in the blood and feeling of the American people more than any other style of music. I believe that it can be made the basis of serious symphonic works of lasting value, in the hands of a composer with talent for both jazz and symphonic music.'

Thirty-five years earlier, Dvorák had addressed the same issue. His three years spent in America gave rise to a String Quartet and a String Quintet both nicknamed 'American', and to the New World Symphony – all superficially inspired by what he took to be Negro music. He said that American concert music could not thrive as an indigenous entity if it slavishly imitated European models. Gershwin put this into focus in one of his most oft-quoted statements: 'Music must reflect the thoughts and aspirations of the people and the time. My people are American. My time is today.'

Once Gershwin made the connection between jazz and the concert-hall (beginning with Rhapsody in Blue), there was no stopping him. It led to a dual career; his concert works span exactly the same years as his Broadway musicals, ending when he got busy on his opera Porgy and Bess. After Rhapsody in Blue came Concerto in F (1925), An American in Paris (1928), Second Rhapsody (1931), Cuban Overture (1932) and Variations on 'I Got Rhythm' (1934). All of these came under attack from critics eager to pounce on structural deficiencies. Gershwin was well aware of his situation – a largely self-taught song-plugger now putting works called 'Rhapsody' or 'Concerto' before the public. 'When my critics tell me now and then I betray a structural weakness,' he said, 'they are not telling me anything I don't know.'

Whatever his technical limitations, Gershwin was the first to put American concert music on the map. The patriotic anthem 'My Country! 'Tis of Thee', or Irving Berlin's 'God Bless America', might have seemed the most appropriate climax for the razzmatazz opening ceremony at the 1984 Olympic Games in Los Angeles. But the spine-chiller proved to be neither of these. It was the synchronous sound of 84 pianists at 84 white pianos playing Rhapsody in Blue.

Gershwin always hankered after tuition from famous musicians. It is hard to see why those he approached – Ravel, Glazunov, Stravinsky, Nadia Boulanger - should take time out to tutor this fast-track product of Tin Pan Alley, with his frenetic lifestyle. His quest led to famous stories (proved to be true). Ravel said: 'Why be a second-rate Ravel when you are already a first-rate Gershwin?' Stravinsky, when told how much Gershwin earned, said: 'Then maybe I should be taking lessons from you.' When he did take lessons from the Americdan Henry Cowell, Gershwin would tart-up an exercise in the style of the 16th-century master of polyphony Palestrina, with what Cowell called 'juicy ninth chords' – as much an essential to Debussy as to Duke Ellington, but definitely out of place in strict counterpoint.

Gershwin's creativity was phenomenal. 'I have more tunes in my head than I could put down on paper in a 100 years,' he said. When he checked out of a hotel and realised he had left behind sketch-books containing about 40 song ideas, he shrugged it off: 'There are plenty more where those came from.' He once said he liked to write six tunes a day to get the bad ones out of his system. In a unique fraternal partnership, his elder brother Ira – already nicknamed 'The Jeweller' because of his polished lyrics for other songwriters – became George's devoted collaborator.

When Ira was asked which came first, the words or the music, he liked to reply: 'The contract.' George made the crafting of their songs sound easy: 'Usually the music comes first. I hit on a new tune, play it for Ira, and he hums it all over the place for a while until he gets an idea for the lyric. Then we work the thing out together.' For Ira it was not so simple. A tune that flowed from George's fingers as he amused himself at the piano might mean days of hard grind. Having memorised it, Ira would often work through the night on numerous drafts to arrive at a preliminary lyric. Then would come stage two. With George at the piano, Ira would set up a bridge table at the top end of the keyboard, spread out his papers, and another Gershwin song would painfully emerge line by line.

Not all their shows were hits, and most of them remain difficult to revive because they were so much part of their time. Their first collaboration, Lady, Be Good! in 1924, helped redefine the Broadway musical. Its musical numbers were more integrated within a story-line than the sort of frothy revues the Gershwins inherited. By 1927 they were offering sharp, political satire in Strike Up The Band – initially a failure but revived in 1930 to great success. Theatre-goers had been through the Wall Street crash and had no illusions about bungling financiers and politicians. Of Thee I Sing (1931), satirising an American presidential campaign, was their greatest triumph. Its jokes about the White House could have come out of today's newspaper.

Gershwin must have been concerned that his songs might not live once the shows had closed. Of the tune in Rhapsody in Blue, he said, 'If I had taken the same themes and put them into songs they would have been gone years ago.' Concert music seemed to have a future, compared with the fragility of Broadway. He need not have worried: so many of his songs are immortal, and jazz performers never tire of spinning improvisations on them.

His crowning achievement, Porgy and Bess, is the most successful American opera of our century. It took a long time to achieve this status, for musical and ethnic reasons. Premiered in 1935, it closed after 124 performances. For an opera, this would be the equivalent of many seasons. But Gershwin mounted it on Broadway, because he wanted 'to develop something that would appeal to the many rather than the cultured few.' Todd Duncan, who created the role of Porgy, remembered Gershwin being 'caught between', believing opera-lovers stayed away because they did not think he could write one, while Broadway thought: 'Georgie's gone high-hat on us.' Gershwin was criticised for producing a hybrid that fluctuated between opera, operetta and musical comedy.

There was also widespread black opposition. The foremost complainer was Duke Ellington who had started his own black opera (called Boola) in 1930, but never completed it. He said that 'no Negro could possibly be fooled by Porgy and Bess'. Nonetheless, the Gershwin borthers and author DuBose Heyward had tried to get away from the whoring, gambling, superstitious black stereotypes portrayed all too often. Fifty years after its premiere, the opera finally reached the stage of the Met. It has since been acclaimed at Glyndebourne and Covent Garden, and numerous centenary year productions around the globe have included a complete performance at the BBC Proms. If only Gershwin had lived to see the triumphant progress of what he called his 'labour of love.'

Gershwin ended his days in Hollywood writing film music. He enjoyed strenuous games of tennis with Schoenberg, who had left Nazi Germany in 1933 and was as fanatical about tennis as he was. He put Schoenberg firmly in his sights as a possible tutor. But it was George's friend, the film star and pianist Oscar Levant, who became a Schoenberg pupil. Levant's nervy, one-movement piano concerto – a schizophrenic mixture of blues, jazz and Alban Berg – is available on CD. It gives a taste of what Gershwin might have written had he tried to follow the same road – although it is most unlikely he could ever have been at home with the Schoenberg aesthetic.

Gershwin died of an undiagnosed brain tumour, 11 weeks short of his 39th birthday. Schoenberg paid simple homage: 'He expressed musical ideas; and they were new – as is the way he expressed them.' The songwriter Harold Arlen said: 'There was nothing phoney about him. He knew he had it and he celebrated it.' Each in his own way, they were defining Gershwin's place in the music of our century.

Rodney Greenberg