  In the first part of this paper, we get new Li-Yau type gradient estimates
for positive solutions of heat equation on Riemmannian manifolds with
$Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack
inequalities are obtained and they lead to new estimates on heat kernels of
manifolds with Ricci curvature bounded from below. In the second part, we
establish a Perelman type Li-Yau-Hamilton differential Harnack inequality for
heat kernels on manifolds with $Ricci(M)\ge -k$, which generalizes a result of
L. Ni \cite{NL1,NL4}. As applications, we obtain new Harnack inequalities and
heat kernel estimates on general manifolds. We also obtain various entropy
monotonicity formulas for all compact Riemannian manifolds.
