  We provide a combinatorial characterization of all testable properties of
$k$-uniform hypergraphs ($k$-graphs for short). Here, a $k$-graph property $P$
is testable if there is a randomized algorithm which makes a bounded number of
edge queries and distinguishes with probability $2/3$ between $k$-graphs that
satisfy $P$ and those that are far from satisfying $P$. For the $2$-graph case,
such a combinatorial characterization was obtained by Alon, Fischer, Newman and
Shapira. Our results for the $k$-graph setting are in contrast to those of
Austin and Tao, who showed that for the somewhat stronger concept of local
repairability, the testability results for graphs do not extend to the
$3$-graph setting. Our proof relies on a random subhypergraph sampling result
proved in a companion paper.
