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We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if 
$X\subseteq \mathbf{P}^{n+1}$ is a very general smooth hypersurface of dimension 
$n$ and degree 
$d\geqslant 2n+1$, then any dominant rational mapping 
$f:X{\dashrightarrow}\mathbf{P}^{n}$ must have degree at least 
$d-1$. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti–Eisenbud concerning varieties of multi-secant lines.
