Monotonicity formulae in geometric analysis have profound applications in many different problems. Quite often these formulae can be derived from Hessian estimates, also known as Li-Yau-Hamilton estimates. These estimates are often called differential Harnack estimates as well, since they imply Harnack estimates by integration along space or spacetime paths. In this talk we will focus on this connection, as well as a novel monotonicity formulae on Einstein manifolds.