Two collatorations at CERN have been pursuing a long‐term goal: to make precision tests of how an atom of antimatter might differ from its ordinary‐matter counterpart. An antihydrogen atom (consisting of a positron bound to an antiproton) might, for example, fall at a slightly different rate than a hydrogen atom. Or the lowest‐lying states of the positron might have slightly different energies than those of the electron in hydrogen. If a precision test indicated that either hypothesis were true, it would shatter a very basic tenet of physics. Different rates of fall would challenge predictions of general relativity, and different energy levels would violate the invariance of physics under the simultaneous operations of charge conjugation, time reversal, and parity inversion.