In the theory of valid inequalities for integer point sets in polyhedra, the traditional, finite-dimensional techniques of polyhedral combinatorics are complemented by infinite-dimensional methods, the study of cut-generating functions. I will give an introduction to these methods and will explain their connection to lattice-free convex bodies. I will present recent results involving inverse semigroups of partial maps, obtained jointly with Robert Hildebrand and Yuan Zhou, and highlight some open questions regarding computability and complexity.