The theory of nonequilibrium fluctuations in open systems is extended to nonlinear situations. It is shown that the usual birth-and-death type of stochastic formulation of chemical kinetics is in general inadequate and has to be replaced by a more detailed phase-space description. As a consequence, for large classes of nonlinear systems arbitrarily far from equilibrium, the classical Einstein fluctuation formula can be extended, provided the steady reference state is asymptotically stable. The case of oscillatory or unstable systems is also discussed. It is conjectured that in such systems, the departure from the steady state is governed by large fluctuations of macroscopic size, while small fluctuations are still described by the extended Einstein formula. Nonequilibrium macroscopic instabilities such as chemical or hydrodynamic instabilities seem therefore to bear strong similarities to first-ordei phase transitions.