  A simplicial complex $H$ consists of a pair of sets $(V,E)$ where $V$ is a
set of vertices and $E\subseteq\mathscr{P}(V)$ is a collection of subsets of
$V$ closed under taking subsets. Given a simplicial complex $F$ and $n\in
\mathbb N$, the extremal number $\text{ex}(n,F)$ is the maximum number of edges
that a simplicial complex on $n$ vertices can have without containing a copy of
$F$. We initiate the systematic study of extremal numbers in this context by
asymptotically determining the extremal numbers of several natural simplicial
complexes. In particular, we asymptotically determine the extremal number of a
simplicial complex for which the extremal example has more than one incomplete
layer.
