  For each fixed $d\ge 1$, we obtain asymptotic estimates for the number of
$d$-representable simplicial complexes on $n$ vertices as a function of $n$.
The case $d=1$ corresponds to counting interval graphs, and we obtain new
results in this well-studied case as well. Our results imply that the
$d$-representable complexes comprise a vanishingly small fraction of
$d$-collapsible complexes.
