By "classical invariants" of manifolds I mean cobordism invariants, and by "quantum invariants" I mean the kind of invariants that come from topological quantum field theories. The methods and structures of algebraic topology give a very complete accounting of cobordism invariants. In fact much of the development of algebraic topology in the twentieth century was inspired by these invariants. In this talk I will discuss the new demands placed on topology by "quantum invariants," and some of the new structures that seem to be relevant. The first half of this talk will be aimed at a general audience. So will the second half, unless time permits, in which case I'll discuss in more detail some of the progress that has been made in this direction.