Around 1974, MacPherson defined a natural transformation between the functor of constructible functions and homology, leading to a functorial theory of Chern classes for possibly singular varieties. These CSM (=Chern-Schwartz-MacPherson) classes, and related notions such as the Chern-Mather class, have been applied in recent years to study questions in a wide variety of fields, ranging from classical enumerative geometry, to birational invariants, to new formulas for Donaldson-Thomas invariants, to computations related to high energy physics. In this talk I will review the theory underlying CSM classes, and survey several of these applications.