He's known as the bloke who does blotches and scribbles - not a particularly sophisticated way of putting it, but as a description it's not bad. Cy Twombly's is hardly a skill that grabs you at first glance. And though we may not particularly expect this from the contemporary, even from works with the most cerebral aesthetic might we not also want a faint stir of feeling? But can you get that from something that looks like a cross between a scientist's notepad and a sink-estate wall?

Apparently you can because this is an artist whose fan base crosses the spectrum from the steely-eyed Tate director Sir Nicholas Serota, who has curated this show, to the impulsive stalker who last year planted a lipstick kiss on one of Twombly's canvases - an act of vandalism that she defended (unsuccessfully) as an expression of sheer love.

But if you - like me - have never felt anything more than a learnt intellectual interest in his work, then the latest Tate Modern exhibition is a must. This is the first major Twombly survey to be staged for 15 years, and maybe you have just seen far too little of this man.

Certainly, the occasional picture strung up in a gallery by a web of art theory isn't enough. You either have simply to accept all those long explanations on trust - and that can be difficult when few artists have attracted such screeds of incomprehensible appreciation. Or you can simply assume that you have just seen the emperor's new clothes.

But now here is a show tracing the course of Twombly's work across almost six decades, from the early 1950s, when he was still a student at the radical Black Mountain college in North Carolina, until today when, having gone in and out of fashion, he is widely respected as one of the world's foremost painters. And as the show progresses, the whole point of this artist moves slowly into view.

Twombly's career does not follow any linear trajectory. His ideas move in great cycles. But think spirals, not circles. Thoughts that curve outwards may slowly come back again, but they never return to exactly the same point. Understandings have moved onwards though they have never lost touch.

The curators of this new show try to capture a sense of this rhythm. Cy Twombly: Cycles and Seasons does not set out to be all encompassing - though it represents a range that extends from the minimalist expanses of his 1970 Treatise on the Veil, which stretches for some 20 paces along one wall, through the lushly impressionistic canvases of an almost baroque series of Venice-inspired paintings, to his lesser known, and far less interesting, sculptures. Instead it presents groups of paintings that have been produced at key moments in his career.

Each group reflects a step in his fundamental search for an artistic language that could reconcile the brash new surfaces of American Abstract Expressionism with the layered traditions of European art history. He is trying to rediscover the relevance of drawing in a world that had erased the graphic line, and find a role for the old gods amid a modernity that had forgotten all about myth. The visitor witnesses a hesitantly experimental artistic voice growing increasingly confident, rising in a dramatic crescendo before suddenly dying away discouraged. You watch an artist pick himself up from criticism, strip his ideas back down to the mainframe of geometry and start again.

His life experiences feed into his work. A period spent working as a cryptographer for the US Army leads to experiments with Surrealist techniques of automatic writing in the calligraphic tangles of his arbitrarily titled first canvases. His first encounter with the ancient world (on a 1957 trip to Italy with Robert Rauschenberg) is captured by inscriptions scratched like half-lost memories on to surfaces pale and worn as old marble. The scribbled calculations that flow across the creamy spaces of his Bolsena paintings directly reflect a fascination with the Apollo 11 flight.

But seldom are links so direct. Twombly has an essentially poetic imagination that gathers inspiration from anywhere from classical stories to American contemporaries. Anyone from Homer to the Marquis de Sade might catch his imagination. The visitor moves through a haze of references. It would take even the most determined academic decades to annotate all the works.

And yet think of T.S.Eliot and his wilfully occluding footnotes to The Wasteland. Twombly is not a painter who wants simply to be explained. You unearth him. It is a gradual, almost physical process. And a show of his work should be treated like an archaeological site. You uncover his canvases like fragments of pottery; you blow at the dust of ideas, stare at shards of half-missing puzzles with half-remembered meanings. A lost world is slowly brought back. It is a pity so many of the canvases are glazed. They should not have that gloss. They should feel as tactile as cave paintings.

Spectators discover something that they recognise, and this can be a fundamentally emotional - almost physical - response. As you stand amid the scatological drama of his 1961 Ferragosta series, for instance, canvases crowding a small gallery in a deliberately claustrophobic hang, you start to feel the sultry oppression of one of those hot days in Rome as the air turns rancid and the rubbish starts to rot. Or look at the 1971 series painted in tribute to a dead friend. The rhythms of scribbles seem almost to embody thought-scrambling grief that turns the inside of the mind into an untuned television set.

Twombly is not about detached intellect. His cryptic scribbles feel a bit like the patterns of a seismograph. A line etched on graph paper can tell you that far away a tremor of the Earth's crust has left thousands of lives in wreckage. It takes time to interpret the knowledge, to let it seep down into the emotional bedrock. But eventually it does.

And yet these feelings cannot be fixed. Mark and emptiness, pattern and disruption, movement and stasis, growth and decay all meet and mingle on his canvases. You stare at them as you stare at the surfaces of his haunting Hero and Leandro triptych. You watch the wave rise and tumble, jumbling the evidence of these doomed lovers' lives, smoothing and settling until nothing but the faintest memory is left floating, a pink trace upon a surface that is as unfixed as water. Twombly is a profound poet. This show left me convinced.

Cy Twombly: Cycles and Seasons, Tate Modern (020-7887 8888), from Thursday, until September 14

The Top Five shows for summer

Vilhelm Hammershøi: The Poetry of Silence

Royal Academy

Discover a world of enchanted serenity in the quietly compelling images of a moody Dane. June 28-September 7, www.royalacademy.org.uk

The Ramayana: Love and Valour in India's Greatest Epic

British Library, London

Vivid illustrations of this Indian epic carry you away to a land of warrior monkeys and wakened giants, dispossesed princes and mighty battles. The story unfurls like a bejewelled comic strip. Until September 14, www.bl.uk

Hadrian: Empire and Conflict

British Museum, London

From his Parthenon to the most tender passion, this show explores the life, love and legacy of Rome's most enigmatic emperor. The show will be housed in the museum's Reading Room, itself originally inspired by the Parthenon. July 24-October 26, www.britishmuseum.org

Tracey Emin: 20 Years

Scottish National Gallery of Modern Art, Edinburgh

The bad girl of Brit Art shows us how to grow up ungracefully in this first retrospective of her career. If you somehow felt you didn't know her before, you will after this. August 2-November 9, www.nationalgalleries.org

Gustav Klimt: Painting, Design and Modern Life in Vienna 1900

Tate Liverpool

The golden boy of the greetings card and poster industry lures you into an era of harmonious design. Until August 31, www.tate.org.uk/liverpool

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