We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations $\kappa = F(\lambda)$ between the principal curvatures $\kappa,\lambda$. In particular we find analytic closed surfaces of genus zero where $F$ is a quadratic polynomial or $F(\lambda) = c\lambda^{2n+1}$. This generalizes results by H.Hopf on the case where $F$ is linear and the case of ellipsoids of revolution where $F(\lambda) = c\lambda^3$.