This paper presents a novel approach to estimating a moving average model of unknown order from an observed time series based on the minimum message length principle (MML). The nature of the exact Fisher information matrix for moving average models leads to problems when used in the standard Wallace-Freeman message length approximation, and this is overcome by utilising the asymptotic form of the information matrix. By exploiting the link between partial autocorrelations and invertible moving average coefficients an efficient procedure for finding the MML moving average coefficient estimates is derived. The MML estimating equations are shown to be free of solutions at the boundary of the invertibility region that result in the troublesome "pile-up" effect in maximum likelihood estimation. Simulations demonstrate the excellent performance of the MML criteria in comparison to standard moving average inference procedures in terms of both parameter estimation and order selection, particularly for small sample sizes.