Euler products: 270 years of history

Report on the Leonhard Euler Festival (St. Petersburg, June 10–12, 2007).
In 1737, Euler published his famous representation of zeta function as a product over primes, thus starting a remarkable series of developments that ranged from number theory (Riemann's zeta and Riemann conjecture) through geometry and theory of modular forms (Hecke operators, l-adic cohomology, Langlands conjecture) to analysis and physics (Selberg's zeta, Witten's program relating geometric Langlands to topological quantum field theory). In the talk, I will present a review of some insights and problems related to this development.