  The "fundamental theorem of Vassiliev invariants" says that every weight
system can be integrated to a knot invariant. We discuss four different
approaches to the proof of this theorem: a topological/combinatorial approach
following M. Hutchings, a geometrical approach following Kontsevich, an
algebraic approach following Drinfel'd's theory of associators, and a physical
approach coming from the Chern-Simons quantum field theory. Each of these
approaches is unsatisfactory in one way or another, and hence we argue that we
still don't really understand the fundamental theorem of Vassiliev invariants.
