The Mumford-Shah functional was introduced in the 1980s as a tool for automatic
image segmentation, but its study gave rise to many interesting questions of
analysis and geometric measure theory. The main object under scrutiny is a free
boundary K where the minimizer may have jumps. The book presents an extensive
description of the known regularity properties of the singular sets K, and the
techniques to get them. It is largely self-contained, and should be accessible
to graduate students in analysis. The core of the book is composed of regularity
results that were proved in the last ten years and which are presented in a more
detailed