  A Gorenstein polytope of index r is a lattice polytope whose r-th dilate is a
reflexive polytope. These objects are of interest in combinatorial commutative
algebra and enumerative combinatorics, and play a crucial role in Batyrev's and
Borisov's computation of Hodge numbers of mirror-symmetric generic Calabi-Yau
complete intersections. In this paper, we report on what is known about smooth
Gorenstein polytopes, i.e., Gorenstein polytopes whose normal fan is
unimodular. We classify d-dimensional smooth Gorenstein polytopes with index
larger than (d+3)/3. Moreover, we use a modification of Oebro's algorithm to
achieve classification results for smooth Gorenstein polytopes in low
dimensions. The first application of these results is a database of all toric
Fano d-folds whose anticanonical divisor is divisible by an integer r larger
than d-8. As a second application we verify that there are only finitely many
families of Calabi-Yau complete intersections of fixed dimension that are
associated to a smooth Gorenstein polytope via the Batyrev-Borisov
construction.
