Classification problems for manifolds can often be framed as identifying a detecting functor from some category of manifolds to an algebraic category. In this talk I will review various successful classification methods and results from this point of view, describing some detecting functors which have so far been discovered. I will then discus how the functorial point of view can help in the classification of manifolds. Disclaimer: the category theory in this talk is at a elementary level. The point is rather to look at examples of categories and functors in the classification of manifolds.