This paper examines robust estimators of core inflation for Belgian historical CPI data, and for euro area Harmonised Indices of Consumer Prices. Evidence of fat tails in the cross-sections of price changes is provided by traditional measures, as well as by a robust measure of the tail weights that is not vulnerable to the masking phenomenon. Trimmed means are considered in the first instance. We introduce a new estimator where the optimal trimming percentage is the lowest percentage for which the hypothesis of normality of the trimmed samples cannot be rejected on the basis of the Jarque-Bera statistic. Two variants are considered, one with a constant and one with a time-varying optimal trimming percentage. The latter has a higher breakdown point. Symmetric and asymmetric trimming are considered as well. Another robust estimator, the one-step Huber-type skipped mean, which is less vulnerable to the masking phenomenon, is also examined. It is shown that the robust estimators outperform the traditional core inflation measures found in the literature. However, as traditional measures, they lag rather than lead observed inflation. This was particularly so in the 70s and the 80s when the oil price shocks had substantial second-round effects on Belgian inflation.