We propose an iterative decomposition that tests and accounts for multiple structural breaks in the mean, seasonality, dynamics and conditional volatility, while also accounting for outliers. Considering each component separately within each iteration leads to greater flexibility compared with joint procedures. Monte Carlo analysis shows the procedure performs well. Applied to monthly CPI inflation in G7 countries and the Euro area, we uncover mean and seasonality breaks for all countries and, allowing for these, changes in persistence are generally also indicated. Further, volatility reductions are widespread in the early to mid 1980s, with some countries exhibiting increases from 1999 onwards.