Euler products: 270 years of history

Доклад на Эйлеровском фестивале (Санкт-Петербург, 10–12 июня 2007 г.).
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.