  K3 polytopes appear in complements of tropical quartic surfaces. They are
dual to regular unimodular central triangulations of reflexive polytopes in the
fourth dilation of the standard tetrahedron. Exploring these combinatorial
objects, we classify K3 polytopes with up to $30$ vertices. Their number is
$36\,297\,333$. We study the singular loci of quartic surfaces that tropicalize
to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant
Theory.
