We present results on the search for two-neutrino double-electron capture (2νECEC) of 124Xe and neutrinoless double-β decay (0νββ) of 136Xe in XENON1T. We consider captures from the K shell up to the N shell in the 2νECEC signal model and measure a total half-life of T2νECEC1/2=(1.1±0.2stat±0.1sys)×1022yr with a 0.87kgyr isotope exposure. The statistical significance of the signal is 7.0σ. We use XENON1T data with 36.16kgyr of 136Xe exposure to search for 0νββ. We find no evidence of a signal and set a lower limit on the half-life of T0νββ1/2>1.2×1024yrat90%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 0νββ. Assuming a 275kgyr 136Xe exposure, the expected sensitivity is T0νββ1/2>2.1×1025yrat90%CL, corresponding to an effective Majorana mass range of ⟨mββ⟩<(0.19–0.59)eV/c2.