Abstract: I will discuss how novel ideas from non-abelian Brill-Noether theory can be used to prove that the moduli space of genus 16 curves is uniruled and that the moduli space of Prym varieties of genus 13 is of general type. For the much studied question of determining the Kodaira dimension of moduli spaces, both these cases were long understood to be crucial in order to make further progress. I will also explain the use of tropical geometry in order to establish the Strong Maximal Rank Conjecture, necessary to carry out this program.