  The traditional concept of phase transitions has, in recent years, been
widened in a number of interesting ways. The concept of a topological phase
transition separating phases with a different ground state topology, rather
than phases of different symmetries, has become a large widely studied field in
its own right. Additionally an analogy between phase transitions, described by
non-analyticities in the derivatives of the free energy, and non-analyticities
which occur in dynamically evolving correlation functions has been drawn. These
are called dynamical phase transitions and one is often now far from the
equilibrium situation. In these short lecture notes we will give a brief
overview of the history of these concepts, focusing in particular on the way in
which dynamical phase transitions themselves can be used to shed light on
topological phase transitions and topological phases. We will go on to focus,
first, on the effect which the topologically protected edge states, which are
one of the interesting consequences of topological phases, have on dynamical
phase transitions. Second we will consider what happens in the experimentally
relevant situations where the system begins either in a thermal state rather
than the ground state, or exchanges particles with an external environment.
