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.