The wave maps equation is an important example of a geometric wave equation, and generalizes the concept of a free wave to manifold-valued maps. There has been significant progress on understanding the dynamics of wave maps from the Minkowski space to both positively and negatively curved targets, especially in the energy critical setting where the spatial domain dimension is two. In this talk I will discuss some of the new behaviors which result from changing the domain geometry. In particular, I will give a survey of recent works with Andrew Lawrie and Sung-Jin Oh in the case where the domain is the hyperbolic space.