A variety is called rational if it is birational to projective space. For example, the only compact, smooth, and rational Riemann surface is the Riemann sphere. A general compact Riemann surface carries two natural invariants which measure its complexity & non-rationality from both a topological and an algebraic perspective: the genus and the gonality. Both of these invariants have classically played a very important role in the study of curves. In higher dimensions there are a number generalizations of these birational invariants which measure the irrationality of an algebraic variety. I will discuss the computation of one of these invariants — the degree of irrationality — and I will pose a number of open problem about these measures of irrationality.