The moduli space of curves M_g is the universal parameter space for
Riemann surfaces of given genus. Its study has been initiated by Riemann
in 1857 and it has been a long-standing problem to describe the nature of
the moduli space as an algebraic variety. I will survey the history of the
problem starting with Severi's conjecture from 1915 predicting that M_g is
always unirational, continuing with the work of Harris and Mumford
spectacularly disproving Severi's conjecture and finally discussing a
recent result which settles this problem in one of the most interesting
remaining cases, that of genus 22.