**Abstract:** The Cheeger problem consists in minimizing the ratio "perimeter over volume" among subsets of a given, bounded domain.
This problem has connections with several variational problems (eigenvalue
estimates, prescribed mean curvature, TV minimization, and others).
After introducing the problem, we will focus on properties of Cheeger sets
(i.e., solutions of the above minimization problem) with special emphasis on
the two dimensional case. In particular, we shall present some recent
results obtained in collaboration with A. Pratelli, on the characterization
of Cheeger sets in non-convex, planar domains.