  We explain new developments in classical knot theory in 3 and 4~dimensions,
i.e. we study knots in 3-space, up to isotopy as well as up to concordance. In
dimension~3 we give a geometric interpretation of the Kontsevich integral
(joint with Jim Conant), and in dimension 4 we introduce new concordance
invariants using von Neumann signatures (joint with Tim Cochran and Kent Orr).
The common geometric feature of our results is the notion of a grope cobordism.
