  We consider the following definition of connectivity in $k$-uniform
hypergraphs: Two $j$-sets are $j$-connected if there is a walk of edges between
them such that two consecutive edges intersect in at least $j$ vertices. We
determine the threshold at which the random $k$-uniform hypergraph with edge
probability $p$ becomes $j$-connected with high probability. We also deduce a
hitting time result for the random hypergraph process -- the hypergraph becomes
$j$-connected at exactly the moment when the last isolated $j$-set disappears.
This generalises well-known results for graphs.
