The Edinburgh International Festival, running from Aug. 10-30, celebrates its 40th birthday this year. Emphasis is on world theater -- companies from Sweden, China, Poland, the United States, Russia and six other countries will perform -- but there are also concerts, operas, exhibitions, talks and seminars, a juggling act, marionette theater, a circus and other offerings almost without number.

Many events are decidedly specialized -- the Stary Theatre of Krakow performing "Crime and Punishment" in Polish, for instance -- but even so tickets for almost everything are difficult to come by. Among the high points will be: Ingmar Bergman directing two plays, August Strindberg's "Miss Julie" (in Swedish) and Henrik Ibsen's "John Gabriel Borkman" (in German). The light opera "Oberon," conducted by Seiji Ozawa. The Toho Company's interpretation of "Medea," directed by Yukio Ninagawa (in Japanese). Tchaikovsky's "Queen of Spades" and "Eugene Onegin," performed by the Maly Theatre of Leningrad.

Performing companies include the Chamber Orchestra of Europe, the Oslo Philharmonic, the Flying Karamazov Brothers, the Brodsky String Quartet, the Chinese Magical Circus, the National Ballet of Canada and the The'a tre de la Salamandre of France. It is truly an international gathering.

You can try for tickets from the festival office at 21 Market St. in Edinburgh or, before leaving home, from Edwards and Edwards, 1 Times Square Plaza, New York, N.Y. 10036, (212) 944-0290. During the festival there is also a booth on The Mound in Edinburgh, where unsold tickets for that day's performances are offered at half price between 1 p.m. and 5 p.m.