Since his death in a Marseille hospital in 1891, aged 37, the French poet and erstwhile enfant terrible Arthur Rimbaud has been virtually canonised as a pioneer of modernism, revered successively by symbolists and decadents, dadaists and surrealists, existentialists and beats, and on into the 1960s when he was a major influence on Bob Dylan and was described by Jim Morrison as "the Master".

Beyond his literary influence he has a kind of mythic status: an existential anti-hero, a "hooligan poet", a nomadic traveller, a rebel against everything including his own literary genius. Not the least remarkable thing about Rimbaud is that his wild, hallucinatory and often very beautiful poems were all written before he turned 21, after which he decisively turned his back on the Bohemian world of Paris and London which had nurtured his talents on a diet of absinthe and hashish - "No more words! I bury the dead in my belly!"

André Breton called Rimbaud a "god of adolescence", and it is in the teenage years that one is most prey to the fatal Rimbaldien charisma. This was the case with Edmund White, who begins his brief biography of the poet with the recollection of how he "discovered" Rimbaud in 1956, when he was 16, in a boarding school near Detroit. "As an unhappy gay adolescent, stifled by boredom and sexual frustration and paralysed by self-hatred, I longed to run away to New York and make my mark as a writer; I identified completely with Rimbaud's desires to be free, to be published, to be sexual, to go to Paris. All I lacked was his courage. And genius." In an interview on his website, however, White adds: "I think I used to identify with Rimbaud and want to be him. Now I think he seems like a horrible brat." He finds him "less heroic as a person but maybe even more interesting as a writer". This is a natural and truthful evolution, though it may also suggest that nearing 70 is not the best age to embark on a biography of Rimbaud.

This is a short book, less than 200 pages of fairly large print, and the absence of any notes or index makes it more an essay than a biography. Or perhaps, given White's distinguished fictional output, one might call it the biographical equivalent of a novella. Its slenderness is further highlighted by the fact that nearly a third of the book covers just five years of Rimbaud's life - the period of his involvement with the older poet Paul Verlaine. This relationship was a seeding-ground for Rimbaud's most challenging poetry - A Season in Hell and Illuminations - but was also a fiasco of drunkenness and savage psychological violence.

Rimbaud barged into Verlaine's life in the late summer of 1870, a 16-year-old country boy from the Ardennes bearing the manuscript of his great poem "Le bateau ivre". A friend of Verlaine's, less than impressed, described him as "a tall, gawky young man, very thin, with the look of a rather fierce street Arab". This is the period of the famous photograph by Etienne Carjat - the tousled hair, the piercing eyes, the skewed necktie, the "strong red mouth with a bitter fold in it". Verlaine was besotted, and deserting his wife and infant son, the pair went on the run to London. They lived in dingy lodgings in Fitzrovia, in a room described by Rimbaud as "full of dirty daylight and the noise of spiders" - the giant plinth of the Post Office Tower now marks the site - and later in Camden Town in a small house on Royal College Street, recently rescued from demolition due to this illustrious connection. Here, piquantly, they advertised in the columns of the Daily Telegraph - "deux gentlemen Parisiens" offering lessons in French and Latin - though whether any would-be pupil ventured unwittingly into this den of demons is not known.

The affair ended with a dreadful showdown in Brussels, where Verlaine shot his tormentor, wounding him in the left wrist. Despite Rimbaud's refusal to press charges, Verlaine was sentenced to two years' hard labour for criminal assault. White writes very well about this relationship, admiring the audacity and defiance of its open homosexuality - though, as he acknowledges, the evidence about Rimbaud's sexuality is contradictory and he cannot overall be enlisted as a gay writer.

After the rupture with Verlaine and the renunciation of literature Rimbaud lived a life of obscure wanderings, glimpsed from time to time as a circus cashier in Hamburg, a quarryman in Cyprus, and a deserter from the Dutch army in Java. He covered huge distances on foot - "I am the tramper of the highway, through the dwarfish woods. The whispering of the sluices covers my steps" ("Childhood"). These punishing excursions were punctuated by spells back home in the Ardennes, but in 1880 he quit Europe for good to go "trafficking in the unknown". The last decade of his life was spent in Aden, on the south coast of Arabia, and in the walled Muslim city of Harar in Ethiopia. He worked as a trader in coffee, skins, cloth, ivory and frankincense. In the mid-1880s he was running guns from the Red Sea port of Tadjourah to the highlands. He returned to France suffering from a tumorous growth on his knee. A swift and harrowing decline followed the amputation of his right leg. His last days of pain and delirium are recorded by his pious and devoted young sister, Isabelle.

The brevity of White's book is no bad thing in itself - length is no guarantee of biographical merit - but there are inevitable sins of omission. This is particularly the case with the later years (nearly half of Rimbaud's life), where he skates over some of the most mysterious and obliquely poetic incidents in Rimbaud's career. White writes in the crisp, laconic style which serves so well in his fiction, but which makes for a rather colourless account of this lurid, difficult and ultimately tragic young poet and adventurer.

• Charles Nicholl's books include Somebody Else: Arthur Rimbaud in Africa (Vintage)