We will study the moduli space of abelian varieties in characteristic $p$ and in particular its supersingular locus $S_g$. We will show when this locus is geometrically irreducible, thereby solving a “class number one problem” or “Gauss problem” for the number of irreducible components; and when a polarised abelian variety is determined by its $p$-divisible group, solving a Gauss problem for central leaves, which are the loci consisting of points whose associated $p$-divisible groups are geometrically isomorphic. Furthermore, we will discuss Oort's conjecture, which states that all generic points of $S_g$ have automorphism group {+/- 1}. This is based on joint works with Ibukiyama and Yu.