We generalize the Castelnuovo bound for the genus of projective curves to higher dimensional varieties and their invariants, such as Betti, Chern and Hodge numbers. The (asymptotically sharp) bounds are given in terms of dimension, codimension and degree. We also bound projective invariants, such as classes and mixed ramification degrees and develop a kind of Morse-Lefschetz theory which suits better the needs of algebraic geometry and allows to bound the number of cells of the CW-complex corresponding to a projective variety in terms of its projective invariants.