  An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of
all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes
generalize the classical hypersimplices as well as cubes, crosspolytopes, and
halfcubes. In this paper we study faces and dissections of $S$-hypersimplices.
Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all
types of multipermutahedra. In analogy to cubes, we also show that the number
of simplices in a pulling triangulation of a halfcube is independent of the
pulling order.
