  Let $K^r_n$ be the complete $r$-uniform hypergraph on $n$ vertices, that is,
the hypergraph whose vertex set is $[n]:=\{1,2,...,n\}$ and whose edge set is
$\binom{[n]}{r}$. We form $G^r(n,p)$ by retaining each edge of $K^r_n$
independently with probability $p$.
  An $r$-uniform hypergraph $H\subseteq G$ is $F$-saturated if $H$ does not
contain any copy of $F$, but any missing edge of $H$ in $G$ creates a copy of
$F$. Furthermore, we say that $H$ is weakly $F$-saturated in $G$ if $H$ does
not contain any copy of $F$, but the missing edges of $H$ in $G$ can be added
back one-by-one, in some order, such that every edge creates a new copy of $F$.
The smallest number of edges in an $F$-saturated hypergraph in $G$ is denoted
by $sat(G,F)$, and in a weakly $F$-saturated hypergraph in $G$ by $wsat(G,F)$.
  In 2017, Kor\'andi and Sudakov initiated the study of saturation in random
graphs, showing that for constant $p$, with high probability
$sat(G(n,p),K_s)=(1+o(1))n\log_{\frac{1}{1-p}}n$, and
$wsat(G(n,p),K_s)=wsat(K_n,K_s)$. Generalising their results, in this paper, we
solve the suturation problem for random hypergraphs for every $2\le r < s$ and
constant $p$.
