In the wake of the Great Recession, many economists began identifying puzzles in inflation dynamics. One is a missing disinflation puzzle, in which inflation in advanced economies in 2008-09 failed to fall as much as expected given the depth of the recession. Another is a missing inflation puzzle, in which inflation has been unexpectedly low in many countries since 2012. These two puzzling episodes have led economists to question and reassess the relation between real activity and inflation and to reconsider the global nature of inflation. As we show in this article (based on Bobeica and Jarociński (2016)), both puzzles are artefacts of models that are too small or too restricted. We show that a rich vector autoregressive (VAR) model explains away the puzzles, and in the process sheds light on the different impacts of domestic real activity and global factors.