  We study eigenvalues of polyharmonic operators on compact Riemannian
manifolds with boundary (possibly empty). In particular, we prove a universal
inequality for the eigenvalues of the polyharmonic operators on compact domains
in a Euclidean space. This inequality controls the $k$th eigenvalue by the
lower eigenvalues, independently of the particular geometry of the domain. Our
inequality is sharper than the known Payne-P\'olya-Weinberg type inequality and
also covers the important Yang inequality on eigenvalues of the Dirichlet
Laplacian. We also prove universal inequalities for the lower order eigenvalues
of the polyharmonic operator on compact domains in a Euclidean space which in
the case of the biharmonic operator and the buckling problem strengthen the
estimates obtained by Ashbaugh. Finally, we prove universal inequalities for
eigenvalues of polyharmonic operators of any order on compact domains in the
sphere.
