  We prove an asymptotic formula for the number of $k$-uniform hypergraphs with
a given degree sequence, for a wide range of parameters. In particular, we find
a formula that is asymptotically equal to the number of $d$-regular $k$-uniform
hypergraphs on $n$ vertices provided that $dn\le c\binom{n}{k}$ for a constant
$c>0$, and $3 \leq k < n^C$ for any $C<1/9.$ Our results relate the degree
sequence of a random $k$-uniform hypergraph to a simple model of nearly
independent binomial random variables, thus extending the recent results for
graphs due to the second and third author.
