  We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex
facet with, say, $d+s$ vertices, called {\it almost simplicial polytopes}. We
provide tight lower and upper bound theorems for these polytopes as functions
of $d,n$ and $s$, thus generalizing the classical Lower Bound Theorem by
Barnette and Upper Bound Theorem by McMullen, which treat the case of $s=0$. We
characterize the minimizers and provide examples of maximizers, for any $d$.
Our construction of maximizers is a generalization of cyclic polytopes, based
on a suitable variation of the moment curve, and is of independent interest.
