The famous knot complement theorem of Gordon and Luecke states that two knots in the three-sphere are equivalent if and only if the complements are homeomorphic. This was proved more than 30 years ago using combinatorial methods. In this talk, we will prove some extended results using techniques from gauge theory. If there is time we will use these methods to settle most of the so-called Cosmetic Surgery Conjecture and another problem from the Kirby Problem List about nullhomotopic knots in arbitrary 3-manifolds. (No knowledge about gauge theory is needed for the talk.) This is joint work with Ali Daemi and Mike Miller Eismeier.