  We discuss the local differential geometry of convex affine spheres in
$\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In
each case, there is a natural metric and cubic differential holomorphic with
respect to the induced conformal structure: these data come from the Blaschke
metric and Pick form for the affine spheres and from the induced metric and
second fundamental form for the minimal Lagrangian surfaces. The local
geometry, at least for main cases of interest, induces a natural frame whose
structure equations arise from the affine Toda system for $\mathfrak
a^{(2)}_2$. We also discuss the global theory and applications to
representations of surface groups and to mirror symmetry.
