  The nuclear recoil effect, known also as the mass shift, is one of
theoretical contributions to the energy levels in muonic atoms. Accurate
theoretical predictions are therefore needed for extracting e.g. the nuclear
charge radii from experimental spectra. We report rigorous QED calculations of
the nuclear recoil correction in muonic atoms, carried out to all orders in the
nuclear binding strength parameter $Z\alpha$ (where $Z$ is the nuclear charge
number and $\alpha$ is the fine structure constant). The calculations show
differences with the previous approximate treatment of this effect, most
pronounced for the lowest-lying bound states. The calculated recoil correction
was found to be sensitive to the nuclear charge radius, which needs to be
accounted for when extracting nuclear parameters from the measured spectra.
