This article proposes a flexible set of transformed polynomial functions for modelling the conditional mean of autoregressive processes. These functions enjoy the same approximation theoretic properties of polynomials and, at the same time, ensure that the process is strictly stationary, is ergodic, has fading memory and has bounded unconditional moments. The consistency and asymptotic normality of the least-squares estimator is easily obtained as a result. A Monte Carlo study provides evidence of good finite sample properties. Applications in empirical time-series modelling, structural economics and structural engineering problems show the usefulness of transformed polynomials in a wide range of settings. © 2014 Wiley Publishing Ltd.