Abstract: After surveying the results of the last 20 years on the structure of effective divisors on $\overline{M}_{0,n}$, we show that the pseudo-effective cone of divisors is not polyhedral for $n\geq8$. Using ideas from Teichmüller dynamics and birational geometry, we construct an extremal non-polyhedral ray of the dual cone of moving curves using residue maps of strata of meromorphic differentials.