  We investigate a family of polytopes introduced by E.M.\ Feichtner, A.\
Postnikov and B.\ Sturmfels, which were named nestohedra. The vertices of these
polytopes may intuitively be understood as constructions of hypergraphs. Limit
cases in this family of polytopes are, on the one end, simplices, and, on the
other end, permutohedra. In between, as notable members one finds associahedra
and cyclohedra. The polytopes in this family are investigated here both as
abstract polytopes and as realized in Euclidean spaces of all finite
dimensions. The later realizations are inspired by J.D.\ Stasheff's and S.\
Shnider's realizations of associahedra. In these realizations, passing from
simplices to permutohedra, via associahedra, cyclohedra and other interesting
polytopes, involves truncating vertices, edges and other faces. The results
presented here reformulate, systematize and extend previously obtained results,
and in particular those concerning polytopes based on constructions of graphs,
which were introduced by M.\ Carr and S.L.\ Devadoss.
