We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality implies all classical Li-Yau type inequalities known to us. Moreover, on a Riemannian manifold, it proves to be equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, and giving new subsequents bounds on the heat kernel of the semigroup.