  The paper considers a manifold $M$ evolving under the Ricci flow and
establishes a series of gradient estimates for positive solutions of the heat
equation on $M$. Among other results, we prove Li-Yau-type inequalities in this
context. We consider both the case where $M$ is a complete manifold without
boundary and the case where $M$ is a compact manifold with boundary.
Applications of our results include Harnack inequalities for the heat equation
on $M$.
