On Poincar\'e series of filtrations

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.