  We show how to construct, starting from a quasi-Hopf algebra, or
quasi-quantum group, invariants of knots and links. In some cases, these
invariants give rise to invariants of the three-manifolds obtained by surgery
along these links. This happens for a finite-dimensional quasi-quantum group,
whose definition involves a finite group $G$, and a 3-cocycle $\om$, which was
first studied by Dijkgraaf, Pasquier and Roche. We treat this example in more
detail, and argue that in this case the invariants agree with the partition
function of the topological field theory of Dijkgraaf and Witten depending on
the same data $G, \,\om$.
