  Equilibrium rate theories play a crucial role in understanding rare, reactive
events. However, they are inapplicable to a range of irreversible processes in
systems driven far from thermodynamic equilibrium like active and biological
matter. Here, we develop a general, computationally efficient nonequilibrium
rate theory in the weak-noise limit based on an instanton approximation to the
stochastic path integral and illustrate its wide range of application in the
study of rare nonequilibrium events. We demonstrate excellent agreement of the
instanton rates with numerically exact results for a particle under a
non-conservative force. We also study phase transitions in an active field
theory. We elucidate how activity alters the stability of the two phases and
their rates of interconversion in a manner that can be well-described by
modifying classical nucleation theory
