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Abstract: Given a smooth projective variety, it is natural to ask (1) How can we determine when it is rational? and (2) If it is not rational, can we measure how far it is from being rational? When the variety is a smooth hypersurface in projective space, these questions have attracted a great deal of attention both classically and recently. An interesting case is when the degree of the hypersurface is at most the dimension of the projective space (the "Fano" range). In joint work with David Stapleton, we show that smooth Fano hypersurfaces of large dimension can have arbitrarily large degrees of irrationality, i.e. they can be arbitrarily far from being rational.