This paper introduces a new model of trend (or underlying) inflation. In contrast to many earlier approaches, which allow for trend inflation to evolve according to a random walk, ours is a bounded model which ensures that trend inflation is constrained to lie in an interval. The bounds of this interval can either be fixed or estimated from the data. Our model also allows for a time-varying degree of persistence in the transitory component of inflation. The bounds placed on trend inflation mean that standard econometric methods for estimating linear Gaussian state space models cannot be used and we develop a posterior simulation algorithm for estimating the bounded trend inflation model. In an empirical exercise with CPI inflation we find the model to work well, yielding more sensible measures of trend inflation and forecasting better than popular alternatives such as the unobserved components stochastic volatility model.