We present a new approach to the approximation of equilibrium solutions to nonlinear rational expectations models that applies to any order of approximation. The approach relies on a particular version of Taylor series approximations - the differential version - and on a scalar perturbation of the support of the entire history of shocks. The resulting solution for any order can always be directly cast in a linear state-space form, permitting the solution to be used for many practical applications such as forecasting, estimation, and computing impulse responses. Using the approach, we show that there cannot be multiple solutions in any order of approximation if the associated first-order approximate solution is determinate. Our approach can be used simply to verify key propositions of the earlier literature, to extend its range of applications, and to resolve puzzles left by it. While the paper only provides an explicit solution up to a third-order approximation, extensions to any higher order approximations are straightforward.