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We study 
$n$-vertex 
$d$-dimensional polytopes with at most one nonsimplex facet with, say, 
$d+s$ vertices, called 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 the Upper Bound Theorem by McMullen, which treat the case where 
$s=0$. 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.
