It has been known since the early 1990s that wall-crossing phenomena are crucial for understanding the spectrum of particles in supersymmetric quantum field theories. More recently, the rich algebraic and geometric structures underlying these phenomena are beginning to be understood. I will give an introduction to this story as it is known today, via examples, focusing mainly on cases related to integrable systems introduced by Hitchin. In this context, the story turns out to be related to the WKB analysis of families of differential equations.