Categorification can be powerful stuff. When Khovanov categorified the Jones polynomial in the late 90's, he obtained a homology theory which was a strictly stronger knot invariant than its underlying knot polynomial. But Khovanov homology can do much more than distinguish knots better. In 2004 Rasmussen showed that Khovanov homology gave an unexpected combinatorial proof of the topological Milnor conjecture. In this talk we provide an introduction to Khovanov homology and background to Rasmussen's result. There will be loads of examples and pictures throughout the talk.