  It is pretty well-known that toric Fano varieties of dimension k with
terminal singularities correspond to convex lattice polytopes P in R^k of
positive finite volume, such that intersection of P and Z^k consists of the
point 0 and vertices of P. Likewise, Q-factorial terminal toric singularities
essentially correspond to lattice simplexes with no lattice points inside or on
the boundary (except the vertices). There have been a lot work, especially in
the last 20 years or so on classification of these objects. The main goal of
this paper is to bring together these and related results, that are currently
scattered in the literature. We also want to emphasize the deep similarity
between the problems of classification of toric Fano varieties and
classification of Q-factorial toric singularities.
