I will describe generalizations, some of which are conjectural, of the canonical ring of a K3 surface to higher dimensional hyper-Kaehler manifolds or to more general Calabi-Yau manifolds. For Calabi-Yau hypersurfaces $X$, for example, I show that the intersection of any two cycles of complementary nonzero dimension is proportional to the canonical 0-cycle (the intersection of a line with $X$). In the hyper-Kaehler case, the canonical ring is generated by the divisor classes and the Chern classes of the tangent bundle and it is conjectured that the cycle class map is injective on it.