 Descriptive set theory is the study of the structural properties of definable sets of real numbers. In this section we shall (1) provide a brief historical overview, (2) introduce the basic notions of descriptive set theory (explaining what we mean by “definable sets” and “structural properties”), (3) describe the central classical results, and (4) discuss the limitative results that ultimately explained why the classical descriptive set theorists had run into difficulties when attempting to generalize their results. 