I will describe some aspects of the breakthrough, due to Vasy in 2010, on global mapping properties of non-elliptic differential operators. In particular, I will explain the process by which one finds Hilbert spaces between which non-elliptic operators like the dAlembertian (the operator in the wave equation) act as a Fredholm maps. The obtaining of Fredholm maps is a central feature of analysis of elliptic operators, and has wide-ranging consequences, for example in index theory and in the analysis of non-linear elliptic PDEs. The current work on the non-elliptic case can be seen as a natural continuation of the geometric microlocal analysis whose use in the elliptic case has been so successful, some of which I will discuss. This is partly joint work with N. Haber and A. Vasy.