  The paper is devoted to highlighting several novel aspects of the moduli
space of curves of genus 13, the first genus g where phenomena related to K3
surfaces no longer govern the birational geometry of M_g. We compute the class
of the non-abelian Brill-Noether divisor on M_13 of curves that have a stable
rank 2 vector bundle with many sections. This provides the first example of an
effective divisor on M_g with slope less than 6+10/g. Earlier work on the Slope
Conjecture suggested that such divisors may not exist. The main geometric
application of our result is a proof that the Prym moduli space of genus 13 is
of general type. Among other things, we also prove the Bertram-Feinberg-Mukai
and the Strong Maximal Rank Conjectures on M_13
