Global developments play an important role for domestic inflation rates. Earlier literature has found that a substantial amount of the variation in a large set of national inflation rates can be explained by a single global factor. However, inflation volatility has been typically neglected, although it is clearly relevant both from a policy point of view and for structural analysis and forecasting. We study the evolution of inflation rates in several countries, using a novel model that allows for commonality in both levels and volatilities, in addition to country-specific components. We find that inflation volatility is indeed important, and a substantial fraction of it can be attributed to a global factor that is also driving inflation levels and their persistence. The extent of commonality among core inflation rates and volatilities is substantially smaller than for overall inflation, which leaves scope for national monetary policies. Finally, we show that the point and density forecasting performance of the model is good relative to standard benchmarks, which provides additional evidence on its reliability.