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.