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