Conze-Lesigne systems are ergodic abelian measure-preserving dynamical systems which are abstractly isomorphic to their second Host-Kra-Ziegler factors. These factors (and their versions of higher order) are relevant in the area of multiple recurrence and related topics in additive combinatorics. In this talk, we present a structure theorem for Conze-Lesigne systems defined for arbitrary actions of a countable discrete abelian group, describing such systems as an inverse limit of translational systems G/L, where G is a locally compact nilpotent group of nilpotency class 2 and L is a lattice in G. Such structure theorems were previously known in the important special cases of finitely generated abelian groups by work of many (in particular, Conze and Lesigne) and direct sums of finite fields by work of Bergelson, Tao, and Ziegler. We will review some of this literature by way of illustrating examples. We then use these examples to help to explain our proof in the general case. If time permits, we present an application of our structure theorem to give a qualitative proof of the inverse theorem for the Gowers U^3-uniformity norm of an arbitrary finite abelian group via a correspondence principle. This talk is based on work jointly with Or Shalom and Terence Tao.