This paper concerns the facial geometry of the set of $n \times n$ correlation matrices. The main result states that almost every set of $r$ vertices generates a simplicial face, provided that $r \leq \sqrt{\mathrm{c} n}$, where $\mathrm{c}$ is an absolute constant. This bound is qualitatively sharp because the set of correlation matrices has no simplicial face generated by more than $\sqrt{2n}$ vertices.