  We use dynamic equations to derive a relation between correlation functions
and response or relaxation functions in many-body systems. The relation is very
general and holds both in equilibrium, when the usual fluctuation-dissipation
theorem is valid, and to linear order in an expansion about arbitrary
non-equilibrium states, when it is not. We illustrate our results by discussing
fluids, both in equilibrium and non-equilibrium states, as well as several
active-matter systems.
