\begin{abstract} A Steiner triple system is a set $A$ together with a collection $S$ of subsets of $A$ of size $3$ such that every two elements of $A$ belong to exactly one element of $S$. They can be treated as a set with a binary operation sending every two different elements $a,b$ to the third element $a\cdot b$ in the subset. If $a=b$, we set $a\cdot b= a$. In this language they are called Steiner quasigroups. We have proved that the theory of Steiner quasigroups has a model completion, a complete theory with $\mathrm{TP}_2$ and $\mathrm{NSOP}_1$ which is not small. \end{abstract}