In this paper, I review the history of "core" PCE inflation and its rationale: remove volatile items with transitory shocks to better highlight the trend in inflation. Structural changes in the inflation process imply that, on a "reducing volatility" basis, the list of items excluded from the "core" inflation basket (aside from gasoline) is far from optimal. This is true whether one assesses volatility on the basis of a weighted component monthly, or an index monthly, or a 12-month index, or a 5-year index. In addition, I demonstrate other deficiencies of exclusion indexes. Excluded items do not just experience transitory shocks, but also have persistent trends; thus excluding them imparts a significant timevarying bias to core inflation. Meanwhile, items that are not excluded can experience volatility and moreover can cause core inflation to depart notably from trend inflation, sometimes at crucial moments. Two other prominent trend inflation measures, trimmed mean PCE inflation and median PCE inflation, gracefully address these issues, but themselves have notable time-varying bias. I discuss the source of the bias in these other measures and how to correct for bias in real time. I then summarize and extend a wide variety of evidence comparing these three trend measures. I conclude that, for a variety of considerations that are relevant for monetary policy deliberations and communication, either trimmed mean PCE inflation or median PCE inflation are superior measures.