A central polynomial of an algebra A is a polynomial f in non-commutative variables that takes central values when evaluated in A. In case it vanishes in A, f is called a polynomial identities of A, otherwise f is a proper central polynomial of A. 

The purpose of this talk is to survey some old and new results on the growth of the spaces of central polynomials, proper central polynomials and polynomial identities of  an algebra  over a field of characteristic zero.