How do the statistical properties of a dynamical system change when the system is perturbed? This is the basic question behind linear response theory. In this talk, I will review the main ideas, motivations, and classical results of the theory, explaining how it provides a rigorous way to predict the effect of small forcings on the long-term statistical behaviour of dynamical systems.

Motivated by complex systems subject to forcings acting on a wide range of time scales, I will then discuss recent results extending the theory to nonautonomous dynamics, where the reference system itself changes with time and invariant measures are replaced by time-dependent equivariant statistical states.