The search for neutrinoless double-electron capture can be a good alternative to the neutrinoless double-beta-decay experiments in shedding light on such aspects of neutrino physics as the neutrino type, non-conservation of the total lepton charge and magnitude of the effective Majorana neutrino mass. The probability of neutrinoless double-electron capture can be resonantly enhanced by some orders of magnitude. Although the phenomenon of the resonant enhancement was predicted some decades ago, the search for resonantly enhanced neutrinoless double-electron-capture transitions was hampered by a lack of precise experimental values of the atomic mass differences of the transition initial and final states. Only recent progress in high-precision Penning-trap mass spectrometry has finally provided suitable means for a determination of atomic masses with a sufficient precision and thus given rise to the experimental campaign for a search for resonantly enhanced transitions. In this article, after an introduction to the theory of neutrinoless double-electron capture and the Penning-trap technique, a review is given on the results of systematic very precise direct Penning-trap measurements of the atomic mass differences of the nuclides which can undergo double-electron capture. The mass differences of 14 nuclear pairs have already been determined with a typical uncertainty of tens to a few hundreds of eV. Partial or even full resonant enhancement of the neutrinoless double-electron-capture probability has been discovered for ¹⁵²Gd and ¹⁵⁶Dy. In ¹⁵⁶Dy a multiple resonance phenomenon has been revealed. For some transitions the half-lives have been estimated (¹⁵²Gd, ¹⁶⁴Er, ¹⁸⁰W, ¹⁵⁶Dy).