Double-beta processes play a key role in the exploration of neutrino and weak interaction properties, and in the searches for effects beyond the standard model. During the last half century many attempts were undertaken to search for double-beta decay with emission of two electrons, especially for its neutrinoless mode 0ν2β−, the latter having still not been observed. Double-electron capture (2EC) was not yet in focus because of its in general lower transition probability. However, the rate of neutrinoless double-electron capture 0ν2EC can experience a resonance enhancement by many orders of magnitude when the initial and final states are energetically degenerate. In the resonant case, the sensitivity of the 0ν2EC process can approach the sensitivity of the 0ν2β− decay in the search for the Majorana mass of neutrinos, right-handed currents, and other new physics. An overview of the main experimental and theoretical results obtained during the last decade in this field is presented. The experimental part outlines search results of 2EC processes and measurements of the decay energies for possible resonant 0ν2EC transitions. An unprecedented precision in the determination of decay energies with Penning traps has allowed one to refine the values of the degeneracy parameter for all previously known near-resonant decays and has reduced the rather large uncertainties in the estimate of the 0ν2EC half-lives. The theoretical part contains an updated analysis of the electron shell effects and an overview of the nuclear-structure models, in which the nuclear matrix elements of the 0ν2EC decays are calculated. One can conclude that the decay probability of 0ν2EC can experience a significant enhancement in several nuclides.