In this thesis, many classical results of topological dynamics are adapted to the set-valued case. In particular, focus is given to the notions of topological entropy and the specification property. These properties are defined in the context of set-valued functions, and examples are given both of classical theorems that extend naturally to this setting and of theorems which have no clear analogue. Particular attention is paid to a result which states that if a dynamical system has the specification property, there exists invariant non-atomic measures with full support.