  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 ($0\nu2\beta^-$), the latter being still not observed.
Double-electron capture (2EC) was not in focus so far because of its in general
lower transition probability. However, the rate of neutrinoless double-electron
capture ($0\nu2$EC) can experience a resonance enhancement by many orders of
magnitude in case the initial and final states are energetically degenerate. In
the resonant case, the sensitivity of the $0\nu2$EC process can approach the
sensitivity of the $0\nu2\beta^-$ decay in the search for the Majorana mass of
neutrinos, right-handed currents, and other new physics. We present an overview
of the main experimental and theoretical results obtained during the last
decade in this field. The experimental part outlines search results of 2EC
processes and measurements of the decay energies for possible resonant
$0\nu$2EC 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 $0\nu2$EC
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 $0\nu2$EC decays are calculated. One can
conclude that the decay probability of $0\nu$2EC can experience a significant
enhancement in several nuclides.
