  A Steiner triple system is a set $S$ together with a collection $\mathcal{B}$
of subsets of $S$ of size 3 such that any two elements of $S$ belong to exactly
one element of $\mathcal{B}$. It is well known that the class of finite Steiner
triple systems has a Fra\"{\i}ss\'e limit $M_{\mathrm{F}}$. Here we show that
the theory $T^\ast_\mathrm{Sq}$ of $M_{\mathrm{F}}$ is the model completion of
the theory of Steiner triple systems. We also prove that $T^\ast_\mathrm{Sq}$
is not small and it has quantifier elimination, $\mathrm{TP}_2$,
$\mathrm{NSOP}_1$, elimination of hyperimaginaries and weak elimination of
imaginaries.
