We present results on the search for two-neutrino double-electron capture (2 nu ECEC) of Xe-124 and neutrinoless double-beta decay (OW) of 136Xe in XENON1T. We consider captures from the K shell up to the N shell in the 2 nu ECEC signal model and measure a total half-life of T-1/2(2 nu EcEc) = (1.1 +/- 0.2(stat) +/- 0.1(sys)) x 10(22) yr with a 0.87 kg yr isotope exposure. The statistical significance of the signal is 7.00. We use XENON1T data with 36.16 kg yr of Xe-136 exposure to search for O nu beta beta. We find no evidence of a signal and set a lower limit on the half-life of T-1/2(beta beta) > 1.2 x 10(24) yr at 90% CL. This is the best result from a dark matter detector without an enriched target to date. We also report projections on the sensitivity of XENONnT to OW. Assuming a 275 kg yr Xe-136 exposure, the expected sensitivity is T-1/2(0 nu) > 2.1 x 10(25) yr at 90% CL, corresponding to an effective Majorana mass range of < m(beta beta)> < (0.19-0.59) eV/c(2).
