  The classical Kruskal-Katona theorem gives a tight upper bound for the size
of an $r$-uniform hypergraph $\mathcal{H}$ as a function of the size of its
shadow. Its stability version was obtained by Keevash who proved that if the
size of $\mathcal{H}$ is close to the maximum, then $\mathcal{H}$ is
structurally close to a complete $r$-uniform hypergraph. We prove similar
stability results for two classes of hypergraphs whose extremal properties have
been investigated by many researchers: the cancellative hypergraphs and
hypergraphs without expansion of cliques.
