It's important to have something to hold on to at the Edinburgh Festival: some sweeping statement that can help ward off panic and cut through the thickets of programme notes about events running all over the city, around the clock. Arrogant dismissal of most of what's on offer is a favoured means of survival; the alternative is utter demoralisation. As ever, the Fringe is unpredictable, amorphous, surprising and infuriating, but the International and Book Festivals obliged by highlighting the broad theme of cultural identity. Although this is a perennial preoccupation in Scotland, this year's political developments have given the debates more urgency. From the composer James McMillan's speech about sectarianism in Scotland at the opening of the Fringe Festival, to last weekend's statement by the Scottish Tourist Board expressing concern that contemporary Scottish cinema is portraying a negative image of the country, the air is thick with questions about Scottish identity.

At the Book Festival, in its cheerful encampment of white tents in Charlotte Square, these questions were aired in daily discussions - on regional identity within Britain, on the Scottish literary tradition, on bilingual writing in Scots Gaelic and English, and even on Alcohol and Cultural Identity. The general conclusion of the last, far-from-sober discussion was that the more opportunities people were given to express themselves and explore their creativity, the less they'd feel the need to turn to drink. Unfortunately the levels of high-spirited alcohol consumption around the city last week seemed to knock that theory on the head. Perhaps the miserably wet, chilly weather was the excuse . . . The International Festival has thrown questions of cultural identity into relief by its joint commission, with the Grec Festival in Barcelona, of two new plays by a young Catalan playwright and a young Scottish playwright, "to mark the opening of the new Scottish parliament", according to the programme. The Speculator by David Greig and The Meeting by Lluisa Cunille were premiered in Catalan at the Grec Festival in July, and then premiered in English by the Traverse Theatre Company at Edinburgh's Royal Lyceum Theatre.

It is no surprise to Irish observers that Scotland should be looking to Catalonia as a model of a flourishing minority culture, with a measure of self-government within a larger European state: we have been down that road already, with the fruitful cross-fertilisation between Els Comedients and Macnas. But the origins of that process were organic; there is an undeniably contrived, programmatic aspect to the Scottish/Catalan cross-commissioning. It's not clear what was gained by the double productions; why not bring the Catalan production of The Meeting to Edinburgh and the English-language production of The Speculator to Barcelona? International festival audiences are well accustomed to watching productions with subtitles.

It's obvious that there are high expectations of the prolific David Greig as the white hope of new Scottish playwriting. His 1996 comic ensemble piece, Caledonia Dreaming, revived for this year's Fringe by the 7:84 company, was hyped as a state-of-the-nation play; also running on the Fringe is his cleverly constructed observation of 1990s relationships, Mainstream - excellently performed by the company he co-founded, Suspect Culture. But his sprawling historical epic, The Speculator, had to carry a freight of expectations about a new history play for a newly independent Scotland. Questions about Scotland's historical relationship with Europe - especially the "auld alliance" with France, and its contribution to European cultural and intellectual life are bounced around by the characters.

Set in Paris in the 1720s, it examines the impact made by the charismatic figure of John Law, the Scottish financier who became fabulously wealthy by persuading Parisians to exchange their gold for paper money and to buy a stake in the great dream of America. Into his ambit come the French playwright, Marivaux, who takes a commission from Law and gambles on a future with an actress, and the young Scot, Lord Islay, who squanders the money Law gives him on the tavern maid he loves.

Drawing on the vaudeville tradition of Scottish theatre and a vein of Scottish writing that examines the condition of exile, The Speculator is an ambitious, three-hour attempt to portray the mood of the period, complete with brawling tavern scenes and Les Miserablesstyle choruses of urchins telling us how chaotic the money markets in Paris were. But despite a valiant effort by David Rintoul to bring substance to the fascinating figure of John Law, the characters remain cardboard cutouts. Philip Hammond's production is cluttered, fussy and slow-moving, and Greig labours his point about the analogy between artistic creation, love and financial speculation - all are leaps of faith that risk everything in pursuit of an imaginative construct.

The Catalan play, The Meeting, also on the vast, proscenium stage of the Royal Lyceum Theatre, demonstrated how broad the two playwrights' brief was. While the two authors were required to reflect the cultural identity of their nations, they had absolute freedom in the pursuit of this theme. The contrast between the two plays surprised a lot of people last week; there were few obvious points of contact between them. Lluisa Cunille certainly didn't provide the Catalan colour and spectacle that might have been expected. In fact her play, The Meeting, has no specific cultural or national references.

It is a quiet, miniaturist piece, which deals with universal themes of isolation, communication and chance. Divided into seven episodes - in which a character emblematically called The Man (John Stahl) encounters a series of characters in railway stations, foyers and on a park bench - it explores the ways in which people manage to connect, or more often in this case, don't connect.

With a delicately lit, minimalist set, this is a low-key production of a restrained, under-stated play which doesn't succeed in holding the attention over an hour-and-a-half. It fails to engage, in part, because it seems so derivative and curiously dated, like an echo of a slew of post-war plays and films, in which urban anomie and existential isolation are played out. So far so disappointing, and the steady stream of walk-outs from The Meeting and the meagre turnout on Saturday to hear the two directors discuss the plays suggested that the risky project of bringing two untried authors to the Lyceum hadn't fired audiences' imagination.

Apart from the Abbey's production, The Wake, which won critical and audience approval, the theatre offerings of the first week of the International Festival had difficulty competing with the Fringe Festival. The competition came mainly from the Traverse Theatre, whose shows are grouped with the Fringe, but in fact inhabit an autonomous zone, consistently showcasing new writing. This year three Irish productions are being staged there: Stones In His Pockets by Marie Jones, which received rave reviews last week, and this week, Rough Magic's The Whisperers and Kabosh's Mojo Mickeybo.

After a couple of days watching some of the more worthy and amateurish efforts on the Fringe, it was a relief to take refuge in the Traverse and to see, among a clutch of impressive productions, Simon Bennett's new play for Max Stafford Clark's Out Of Joint company, Drummers. Despite being Bennett's debut, every aspect of this production - script, performances, pacing, set design, and lighting - bore the hallmarks of a highly experienced director. Depicting the state of mind of a newly-released prisoner from Brixton who returns to his former way of life of house burglary while his younger brother has become a heroin addict, it is distinguished from other plays on these themes by its sense of authenticity, which derives from the playwright's experience of burglary and prison. With an obvious debt to Mark Ravenhill's Shopping and F--king, this is a tough, violent, sometimes shocking work, which can make the audience feel like uncomfortable voyeurs, but its bleakness is illuminated by two stunning performances by Peter Sullivan as the released prisoner and Maggie McCarthy as his mother.

For respite from the grinding cycle of drugs, crime and prison, which featured heavily on the Fringe last week, audiences were more than happy to take a holiday to the Greek island of Cephalonia where Louis de Bernieres's best-selling novel, Captain Corelli's Mandolin, is set. Crammed into an upper room in Edinburgh's favourite Italian delicatessen, Valvona & Crolla, amid the aromas of cheese, bread and olives and with beakers of wine in hand, they laughed and wept (really) their way through a delicate twohanded adaptation of the novel, performed by Philip Contini and Mike Maran in the style of a work for children, with a minimum of simple, cut-out props.

Beautifully accompanied by Allison Stephens on mandolin and Anne Evans on piano, the two able raconteurs create the cast of island characters and their invaders and tenderly elaborate the Greek-Italian love story at the centre of the novel. Greek - both modern and ancient - French, and English emphasised the enduring resonance of the Antigone myth in a production of Sophocles's Antigone that was remarkable for its stark simplicity. This was a no-budget Fringe production, performed at 10.30 a.m. with total commitment by Tomee Theatre from Athens, making highly effective, visually dramatic use of a few props and some choreographed ensemble sequences. With its incorporation of song and music, it was an intensely engaging adaptation. At the opposite end of the scale was a piece of cross-cultural borrowing which elicited warm audience reaction at the Edinburgh Playhouse and excoriating critical comment: a lavish Japanese production (from Tokyo's Bunkamura Arts Centre) of Puccini's late foray into chinoiserie: Turandot. Saburo Teshigawara was the director, production designer, lighting and costume designer and choreographer of this production - which probably didn't make him much fun to be around but certainly imposed a consistent visual style: consistently busy.

This was arena opera at its most flashy, with ever more flamboyant stage images to match almost every phrase of music, as if Teshigawara were reluctant simply to let the singers stand and sing. Judging by the choice of principals and the way the dramatic moments of the opera were deliberately played down or thrown away, Teshigawara is not actually very interested in opera.

There were highly effective moments of vivid spectacle, some arresting dance inserts and beautifully choreographed choral sequences, but the facile use of video images of violence and urban bleakness, and the general overloading of stage activity threatened to sink the entire enterprise. Worst of all, though, was the fact that Turandot (Chieko Shimohara) mangled Puccini's lilting musical phrasing. She scraped and scooped tortuously through the vast aria, In Questa Reggia, until it became incomprehensible why her suitor, Calaf, should want to spend a minute in her presence, let alone risk execution in his attempt to win her hand. It was disappointing that Turandot was the only opera on offer in the first week of the International Festival; the emphasis in the programme was on contemporary dance, with the Dutch National Ballet topping the bill with a magnificent display of technical brilliance and theatrical effects in William Forsyth's lyrical, full-length, abstract ballet, Artefact.

Less distinctive, and considerably more self-regarding, was the series of short pieces by the young French choreographer, Boris Charmatz. They included a promenade production, Aatt Enen Tionon, in which three dancers are isolated on three levels of a scaffolded set. Wearing white Tshirts and no underwear (all fur coats and no knickers, almost), they thrash about in their allotted spaces for 35 minutes, like insects trapped in a jar. It's an intense, self-conscious image of claustrophobia. Also employing a variation on the Eurocrash choreographic style, the (highly capable) dancers in Meg Stuart's Brussels-based Damaged Goods company hurl themselves with destructive ferocity onto the floor in Appetite, scrambling to their feet and collapsing again, as if constantly thwarted by gravity. Never allowed to soar or to flow, uninterrupted movement is impossible. This is a choreographic cul de sac, in many ways, which wasn't helped by the desultory, obscure linking sequences, which, presumably were the contribution of Stuart's visual collaborator for this piece, the installation artist, Ann Hamilton. At an-hour-and-a-half, Appetite cried out for editing and in its self-indulgence it belonged more comfortably on the Fringe - in a cheaper production. But then again, it's obvious that, apart from questions of scale, the distinctions between the two festivals are becoming increasingly blurred and are not very useful: the only thing to do is to forget about preconceived categories and identifiable themes. Just bring an open mind and plunge in. Luckily, there are two more weeks in which to do just that.

For the Fringe Festival, telephone: 0044 131-226 5138.

For the International Festival, telephone: 0044 131-473 2000.