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We study the enumerative geometry of the moduli space ${ℛ}_g$ of Prym varieties of dimension $g-1$. Our main result is that the compactication of ${ℛ}_g$ is of general type as soon as $g>13$ and $g$ is different from 15. We achieve this by computing the class of two types of cycles on ${ℛ}_g$: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym–Green conjecture on syzygies of Prym-canonical curves....