  In this survey article we discuss the interior and boundary regularity of
Alexandrov solutions to $\det D^2u = 1$. We include some topics which it seems
were not recently revisited in similar articles, including Calabi's interior
$C^3$ estimate, and the approaches of Cheng-Yau and Lions to obtain classical
solutions to the Dirichlet problem.
  The survey grew from two mini-courses given by the author in May 2018. One
was for "Advanced Lectures in Nonlinear Analysis" at l'Universit\`{a} degli
Studi di Torino, and the other for the Oxford PDE CDT.
