The Darboux theory of integrability for planar polynomial di®erential equations is a classical ¯eld, with connections to Lie symmetries, di®erential algebra and other areas of mathematics. In the present paper we introduce the concepts and problems, give an overview of some recent results on inverse problems, and prove a general ¯niteness result for the case of prescribed integrating factors. We also include a number of new and relevant examples and applications.