Ricci flow is a quasilinear parabolic PDE that evolves a Riemannian metric on a manifold according to its Ricci curvature. The fixed points of suitably normalized Ricci flow include Einstein metrics and, more generally, Ricci solitons. Analysing the stability of these fixed points is important to understanding the long-time behaviour of Ricci flow, and is also interesting from viewpoint of dynamical system. In this talk, we will survey the results on linear and dynamical stability of Einstein metrics and Ricci solitons under Ricci flow. In particular, we will explain how to set up the appropriate interpolation spaces and use the maximal regularity theory to deduce dynamical stability from linear stability.