As inflation targeting (IT) turns 35, it has become a key institutional monetary framework by central banks. Yet, this paper shows that stark differences exist among inflation targeting countries in the conduct of monetary policy. Behind such heterogeneity, the legacy of a high inflation history appears as a preponderant factor. We propose a model that diverges from existing IT workhorse models by adding path-dependence (to a forward-looking model) and potentially imperfect central bank credibility. We show that achieving low inflation (hitting the target) requires more aggressive monetary policy, and is costlier from an output point of view, when individuals’ past inflationary experiences shape their inflation expectation formation. In turn, we provide empirical evidence of the need for these two theoretical additions. Countries that experienced a high level of inflation before adopting the IT regime tend to respond more aggressively to deviations of inflation expectations from the central bank’s target. We also point to the existence of a credibility puzzle, whereby the strength of a central bank’s monetary policy response to deviations from the inflation target remains broadly unchanged even as central banks gain credibility over time. Put differently, a country’s inflationary past casts a long and persistent shadow on central banks.