This thesis presents some new results on Dehn surgery. The overarching theme of the thesis is to find restrictions on obtaining a 3-manifold by a Dehn surgery on a knot in another 3-manifold (although we also find new examples in chapter 5) and most of these restrictions are obtained by exploring the consequences of the mapping cone formula in Heegaard Floer homology.

In particular, we show that only finitely many alternating knots can yield a given 3-manifold by Dehn surgery and confirm the knot complement conjecture for many classes of knots.