A proven method for obtaining statistical characterisations of sufficiently chaotic dynamical systems is to study the spectral properties of the associated Perron-Frobenius operator. Such characterisations usually take the form of a statistical law e.g. a central limit theorem or large deviation principle. It is natural to ask if these statistical laws, and their parameters, are robust to perturbations in the dynamics, which may arise e.g. via the idealisation of a physical system, or the numerical approximation of an abstract one. We will review the existing theory on the stability of invariant measures, as well as recent work by C. and Froyland on the stability of more sophisticated statistical descriptions of dynamical systems.