For any fixed rational number p/q, Dehn surgery gives a map from the set of knots in the 3-sphere to the set of closed orientable 3-manifolds. In 1978, Gordon conjectured that these maps are never injective. I will discuss some results which demonstrate non-injectivity for some special cases of p/q, before going on to discuss joint work with Kyle Hayden and Lisa Piccirillo in which we prove the conjecture.