Abstract: Homology is easy to compute, thanks to excision, but it isn't very sensitive. It only detects homotopy types. In this talk I'd like to give one answer to the question: Is there a notion of homology theory for manifolds that's sensitive to more? I will present the definition of factorization homology, which Lurie has also called topological chiral homology. Factorization homology generalizes usual Eilenberg-Steenrod homology, and is and invariant of manifolds and stratifications on them. The main result will be a classification of all homology theories, namely by giving an equivalence between the category of homology theories and the category of certain kinds of algebras. I will explain how the theorem in turn gives candidates for new sources of invariants of embedding spaces (and in particular, link invariants). If time allows, I can discuss connections to topological field theories and to Koszul duality. This is joint work with David Ayala and John Francis.