The famous Gordon-Luecke knot complement theorem says that knots in the three-sphere are determined by their complements. This was settled by proving that non-trivial knots in the three-sphere cannot have non-trivial Dehn surgeries to the three-sphere. More generally, the Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss some results towards cosmetic surgery and knot complement problems. This is joint work with Ali Daemi and Mike Miller Eismeier.