If the 3-manifold theory is understood broadly enough, then one should mention the Vassiliev conjecture and in general, the problem of computing the cohomology of the spaces of knots in 3-manifolds. Note that for manifolds of dimension 4 or more this has been completely solved. For an introduction to all this see Vassiliev's ICM 1994 talk (MR1403923MR1403923) and for more details see his Complements of discriminants of smooth maps (MR1168473MR1168473).