We establish universal
inequalities for the eigenvalues of the clamped plate problem on compact
submanifolds of Euclidean space, of spheres and of real, complex and quaternionic
projective spaces. We prove similar results for the biharmonic operator on
domains of Riemannian manifolds that admit spherical eigenmaps (this includes
compact homogeneous Riemannian spaces) and finally on domains of hyperbolic
space.