We investigate changes in the time series characteristics of postwar U.S. inflation. We model the conditional mean of inflation with a long memory autoregressive fractionally integrated moving average model allowing for stochastic volatility in the conditional vari- ance of the process. We apply exact maximum likelihood to monthly data to get efficient estimates of the parameters in subsamples of varying size. By this newly developed model and estimation technique we find remarkable changes in the variance, in the order of in- tegration, in the short memory characteristics and in the volatility of volatility. Monetary authorities, financial institutions, pension funds and private investors demand re- alistic statistical models for inflation to assess the real value of wealth, income and returns, both in the short run and in the long run. In this paper we develop a new exact maximum likelihood method to investigate the statistical properties of an inflation process with long memory and stochastic volatility. It is well established that the statistical properties of postwar U.S. inflation underwent a number of structural breaks. There are many ways to explain persistent changes in the mean, variance and autocorrelation of U.S. inflation. The changes in the time series properties of inflation may be due to changing monetary policies (short run direct price controls in the 1950s and 1970s, long run indirect inflation controls starting in the 1980s) or to changes in the process generating price shocks. Many types of shocks have been investigated, we mention technological progress, unemployment changes, output gap disturbances, fluctuations in real unit labour costs as in Gal´o and Gertler (1999), oil price shocks, as in Hooker (1999), changes in the sectoral distribution of price changes as in Ball and Mankiw (1995), trade unions as in Bowdler and Nunziata (2007), and exchange rate pass-through as in Campa and Goldberg