Let \(\pi: M\to B\) be a Riemannian orbifold submersion with totally geodesic leaves such that for any \(V\in T_x F\) (\(F\) is the leaf) and any \(X,Y\in T_x M\) it holds that \(R(X,Y)V=\langle Y,V\rangle X-\langle X,V\rangle Y\) for each \(x\in M\). Then \(M\) is said to be \(n\)-Sasakian, where \(n=\dim F\). This is a generalization of the notion of a 3-Sasakian manifold [cf. \textit{C. Boyer} and \textit{K. Galicki}, Surv. Differ. Geom., Suppl. J. Differ. Geom. 6, 123--184 (1999; Zbl 1008.53047)]. These manifolds have many remarkable geometric properties which should be compared with those of 3-Sasakian manifolds. The author furnishes examples of \(n\)-Sasakian manifolds and makes links to the isoparametric hypersurfaces. The examples carry Einstein metrics.