This paper considers the application of long-memory processes to describing inflation for ten countries. We implement a new procedure to obtain approximate maximum likelihood estimates of an ARFIMA-GARCH process; which is fractionally integrated I(d) with a superimposed stationary ARMA component in its conditional mean. Additionally, this long memory process is allowed to have GARCH type conditional heteroscedasticity. On analysing monthly post-World War II CPI inflation for ten different countries, we find strong evidence of long memory with mean reverting behaviour for all countries except Japan, which appears stationary. For three high inflation economies there is evidence that the mean and volatility of inflation interact in a way that is consistent with the Friedman hypothesis.