  We discuss topics on the geometry of the moduli space of curves. We present a
short proof of the Harris-Mumford theorem on the Kodaira dimension of the
moduli space which replaces the computations on the stack of admissible covers
by a simple study of tautological Koszul bundles on M_g. We also present a
streamlined self-contained account of Verra's recent work on the unirationality
of M_g. Finally, we discuss a proof that the moduli space of curves of genus 22
is of general type. Written for Surveys in Differential Geometry.
