This set of notes is an introductory survey on box spaces and their connections with Riemannian and spectral geometry. After defining box spaces of finitely generated residually finite groups $G$, we give a brief introduction to coarse geometry, then the link between box spaces and expanders, and the dictionary between coarse geometric properties of box spaces and analytic properties of $G$. When $G$ is the fundamental group of a closed Riemannian manifold $M$, there are deep connections between the geometry of a box space of $G$ and the geometry of the associated tower of Riemannian covers of $M$.