  In this survey one discusses the notion of the Poincar\'e series of
multi-index filtrations, an alternative approach to the definition, a method of
computation of the Poincar\'e series based on the notion of integration with
respect to the Euler characteristic (or rather on an infinite-dimensional
version of it), generalizations of the notion of the multi-variable Poincar\'e
series based on the notion of the motivic integration, and relations of the
latter ones with some zeta functions over finite fields and with generating
series of Heegaard-Floer homologies of algebraic links.
