We will discuss how the monoid of knotted circles in 3-space can be filtered by a family of equivalence relations involving "gropes". Gropes are 2-complexes that are constructed inductively from surfaces glued together. In 3-dimensions, this filtration is related to the Kontsevitch integral. Gropes can also be used in a 4-dimensional setting to filter the knot concordance group. We will explain how noncommutative algebra and analysis yield new information about this filtration. The work described is joint with Peter Teichner.