  A well-known principle of Mumford asserts that all moduli spaces of curves
are varieties of general type, except a finite number of cases that occur for
relatively small genus, when these varieties tend to be uniruled. In all known
cases, the transition from uniruledness to being of general type is quite
sudden. For the first time, we show that naturally defined moduli spaces of
curves can have intermediate birational type. We show that the moduli space of
even spin curves of genus 8 is of Calabi-Yau type. Then we prove that the
41-dimensional moduli space M_{11, 11} of 11-pointed curves of genus 11, has
Kodaira dimension 19.
