Enumerative geometry is concerned with counts of geometric objects satisfying given constraints. In algebraic geometry, these can be related to geometric spaces, such as curves, and objects of more algebraic flavor, like sheaves, which encode equations and are a generalization of vector bundles. Under certain assumptions, one hopes to consider a moduli space that parametrizes the objects in question and then obtain numerical invariants by integrating cohomology classes capturing the constraints against an appropriate fundamental class in its homology. In this talk, we will review sheaf counting and then discuss how to extend this approach to cases of interest where the usual assumptions do not hold. An important application motivated by physics is the construction of new generalized Donaldson-Thomas invariants enumerating sheaves on Calabi-Yau threefolds.