  In this expository paper we describe an unifying approach for many known
entropies in Mathematics. First we recall the notion of semigroup entropy h_S
in the category S of normed semigroups and contractive homomorphisms, recalling
also its properties. For a specific category X and a functor F from X to S, we
have the entropy h_F, defined by the composition of h_S with F, which
automatically satisfies the same properties proved for h_S. This general scheme
permits to obtain many of the known entropies as h_F, for appropriately chosen
categories X and functors F. In the last part we recall the definition and the
fundamental properties of the algebraic entropy for group endomorphisms, noting
how its deeper properties depend on the specific setting. Finally we discuss
the notion of growth for flows of groups, comparing it with the classical
notion of growth for finitely generated groups.
