We generalize the Lp spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. We show that these bounds are optimal on any manifold in a very strong sense. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators and their optimality follows by semi-classical analysis. The talk is based on joint work with Julien Sabin.