The gonality of a smooth projective curve is the smalles degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. I will discuss some of these notions, and present joint work with L. Ein and R. Lazarsfeld. Our main rezult is thet if X is an n-dimensional hypersurface of degree d at least (5/2)n, then any dominant rational map from X to P^n must have degree at least d-1.