In this informal talk, I will introduce some notions from derived algebraic geometry, an emerging discipline combining homotopy theory and algebraic geometry. In particular I will discuss derived loop spaces, which give geometric interpretations of Hochschild and cyclic homology. My goal is to describe applications of these loop spaces to representation theory and topological field theory, through their relations to D-modules and Drinfeld doubles (joint work with David Nadler and John Francis). I will not assume technical familiarity with any of these topics.