We will report on some recent results concerning the topological entropy and dynamics of continuous free semigroup actions. We give sufficient topological conditions for the topological entropy to be positive and to be given by the exponential growth rate of “non-autonomous” periodic points. In the case of semigroups of Ruelle-expanding maps,
parameterized by random walks on the free semigroup, we will discuss the regularity of the pressure function, establish the main properties of the dynamical zeta function and prove the existence of stationary probability measures. This is a joint work with M. Carvalho (Porto) and F. Rodrigues (Universidade Federal do Rio Grande do Sul).