The author proves the following theorems: (1) Let \({\mathcal F}\) be a Riemannian foliation of a compact manifold \(M\) with constant curvature \(\kappa\). If \(\kappa=0\), then \(M\) splits locally isometrically as \(B \times F\) and the leaves of \({\mathcal F}\) locally coincide with \(\{p\} \times F\), \(p \in B\). If \(\kappa<0\), then no such foliation exists. (2) A compact locally symmetric space of negative curvature admits no Riemannian foliations.