  The characterization of lattice polytopes based upon information about their
Ehrhart $h^*$-polynomials is a difficult open problem. In this paper, we finish
the classification of lattice polytopes whose $h^*$-polynomials satisfy two
properties: they are palindromic (so the polytope is Gorenstein) and they
consist of precisely three terms. This extends the classification of Gorenstein
polytopes of degree two due to Batyrev and Juny. The proof relies on the recent
characterization of Batyrev and Hofscheier of empty lattice simplices whose
$h^*$-polynomials have precisely two terms. Putting our theorem in perspective,
we give a summary of these and other existing results in this area.
