In this paper we discuss several features of Hilbert spaces of holomorphic functions on domains of Cn. We study composition and multiplication operators on generalized Bergman spaces and give results on the dynamical behaviour (i.e.cyclicity, hypercyclicity, compactness) of the first ones and on the algebraic properties of the space that the second one interprets. In particular we underline the analogies and differences between the case of bounded and unbounded domains in C and Cn. © 2004 American Institute of Physics