  The past decade has witnessed the development of systematic effective
theories for dissipative thermal systems. Here, we describe an analogous
effective theory framework that applies to the classical stochastic dynamics of
nonequilibrium systems. We illustrate this approach using a range of examples,
including nonreciprocal (predator-prey) dynamics, dissipative and driven
rigid-body motion, and active chiral fluids and solids. Many of these systems
exhibit a generalized time-reversal symmetry, which plays a crucial role within
our formalism, and in many cases can be implemented within the
Martin-Siggia-Rose path integral. This effective theory formalism yields
generalizations of the fluctuation-dissipation theorem and second law of
thermodynamics valid out of equilibrium. By stipulating a stationary
distribution and a set of symmetries -- rather than postulating the stochastic
equations of motion directly -- this formalism provides an alternative route to
building phenomenological models of driven and active matter. We hope that this
approach facilitates a systematic investigation of the universality classes of
active matter, and provides a common language for nonequilibrium many-body
physics from high energy to condensed matter.
