A set W of positive measure in R^d is said to be spectral if the space L^2(W) has an orthogonal basis of exponential functions. A conjecture of Fuglede (1974) stated that W is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body W the "tiling implies spectral" part of the conjecture is true. In this talk we will sketch the proof of the other direction of Fuglede's conjecture for convex bodies, that is, any spectral convex body tiles the space by translation. Joint work by NIr Lev and Mate Matolcsi.