  For a positive integer $n$, a graph with at least $n$ vertices is
$n$-existentially closed or simply $n$-e.c. if for any set of vertices $S$ of
size $n$ and any set $T\subseteq S$, there is a vertex $x\not\in S$ adjacent to
each vertex of $T$ and no vertex of $S\setminus T$. We extend this concept to
uniform hypergraphs, find necessary conditions for $n$-e.c. hypergraphs to
exist, and prove that random uniform hypergraphs are asymptotically
$n$-existentially closed. We then provide constructions to generate infinitely
many examples of $n$-e.c. hypergraphs. In particular, these constructions use
certain combinatorial designs as ingredients, adding to the ever-growing list
of applications of designs.
