  This is an expository article presenting various results about the Cheeger
problem and its connections with other relevant mathematical problems. In
section 2 we synthetically describe three mathematical problems (eigenvalue
estimates, prescribed mean curvature equation, ROF model for image
segmentation) focusing on their link with the Cheeger problem. Then, a
selection of classical as well as more recent results obtained in collaboration
with Aldo Pratelli will be described in the remaining sections.
