Abstract: Given an algebraic singularity, there are two fundamentally different kinds of homological invariants that can be assigned to it: first, it is possible to resolve the singularity and construct invariants from such a resolution via a gluing process; the resulting invariants are topological-geometric in nature. Second, there are algebraic invariants that can not be obtained in this fashion. I will discuss singularities, resolutions, and examples of both kinds of invariants.