I will survey the ways in which manifold calculus of
functors has in recent years been used for extracting information
about the topology of spaces of knots and links. Related
homotopy-theoretic techniques, such as cosimplicial spaces and
operads, will also be mentioned along the way. I will end with some
recent results about spaces of homotopy string links and in particular
about how one can use functor calculus to extract information about
Milnor invariants.