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An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containingany member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ bethe collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edgessuch that for some $\left(\ell+1\right)$-set $S$ every pair $\{u, v\} \subsetS$ is covered by an edge in $F$. Our first result shows that for each $\ell\geq r \geq 2$ every $\mathcal{K}_{\ell+1}^{r}$-saturated $r$-graph on $n$vertices with $t_{r}(n, \ell) - o(n^{r-1+1/\ell})$ edges contains a complete$\ell$-partite subgraph on $(1-o(1))n$ vertices, which extends a stabilitytheorem for $K_{\ell+1}$-saturated graphs given by Popielarz, Sahasrabudhe andSnyder. We also show that the bound is best possible. Our second result ismotivated by a celebrated theorem of Andr\'{a}sfai, Erd\H{o}s and S\'{o}s whichstates that for $\ell \geq 2$ every $K_{\ell+1}$-free graph $G$ on $n$ verticeswith minimum degree $\delta(G) > \frac{3\ell-4}{3\ell-1}n$ is $\ell$-partite.We give a hypergraph version of it. The \emph{minimum positive co-degree} of an$r$-graph $\mathcal{H}$, denoted by $\delta_{r-1}^{+}(\mathcal{H})$, is themaximum $k$ such that if $S$ is an $(r-1)$-set contained in a edge of$\mathcal{H}$, then $S$ is contained in at least $k$ distinct edges of$\mathcal{H}$. Let $\ell\ge 3$ be an integer and $\mathcal{H}$ be a$\mathcal{K}_{\ell+1}^3$-saturated $3$-graph on $n$ vertices. We prove that ifeither $\ell \ge 4$ and $\delta_{2}^{+}(\mathcal{H}) >\frac{3\ell-7}{3\ell-1}n$; or $\ell = 3$ and $\delta_{2}^{+}(\mathcal{H}) >2n/7$, then $\mathcal{H}$ is $\ell$-partite; and the bound is best possible.This is the first stability result on minimum positive co-degree forhypergraphs.

Two stability theorems for $\mathcal{K}_{\ell + 1}^{r}$-saturated hypergraphs