Topological and metric entropy for group and semigroup actions

It is well-known that the classical definition of topological entropy for group and semigroup actions is frequently zero in some rather interesting situations, e.g. smooth actions of ℤk+ (k >1) on manifolds. Different definitions have been considered by several authors. In the present article we describe the one proposed in 1995 by K.H.Hofmann and the author which produces topological entropy not trivially zero for such smooth actions. We discuss this particular approach, and also some of the main properties of the topological entropy defined in this way, its advantages and disadvantages compared with the classical definition. We also discuss some recent results, obtained jointly with Andrzej Biś, Dikran Dikranjan and Anna Giordano Bruno, of a similar definition of metric entropy, i.e. entropy with respect to an invariant measure for a group or a semigroup action, and some of its properties.