  There are open questions about which Newton polygons and Ekedahl-Oort types
occur for Jacobians of smooth curves of genus $g$ in positive characteristic
$p$. In this chapter, I survey the current state of knowledge about these
questions. I include a new result, joint with Karemaker, which verifies, for
each odd prime $p$, that there exist supersingular curves of genus $g$ defined
over $\overline{{\mathbb F}}_p$ for infinitely many new values of $g$. I sketch
a proof of Faber and Van der Geer's theorem about the geometry of the $p$-rank
stratification of the moduli space of curves. The chapter ends with a new
theorem, in which I prove that questions about the geometry of the Newton
polygon and Ekedahl-Oort strata can be reduced to the case of $p$-rank $0$.
