  We present results on the search for double-electron capture
($2\nu\text{ECEC}$) of $^{124}$Xe and neutrinoless double-$\beta$ decay
($0\nu\beta\beta$) of $^{136}$Xe in XENON1T. We consider captures from the K-
up to the N-shell in the $2\nu\text{ECEC}$ signal model and measure a total
half-life of
$T_{1/2}^{2\nu\text{ECEC}}=(1.1\pm0.2_\text{stat}\pm0.1_\text{sys})\times
10^{22}\;\text{yr}$ with a $0.87\;\text{kg}\times\text{yr}$ isotope exposure.
The statistical significance of the signal is $7.0\,\sigma$. We use XENON1T
data with $36.16\;\text{kg}\times\text{yr}$ of $^{136}$Xe exposure to search
for $0\nu\beta\beta$. We find no evidence of a signal and set a lower limit on
the half-life of $T_{1/2}^{0\nu\beta\beta} > 1.2 \times 10^{24}\;\text{yr}\;
\text{at}\; 90\,\%\;\text{CL}$. 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 $0\nu\beta\beta$. Assuming a
$275\;\text{kg}\times\text{yr}$ $^{136}$Xe exposure, the expected sensitivity
is $T_{1/2}^{0\nu\beta\beta} > 2.1 \times 10^{25}\;\text{yr}\; \text{at}\;
90\,\%\;\text{CL}$, corresponding to an effective Majorana mass range of
$\langle m_{\beta\beta} \rangle < (0.19 - 0.59)\;\text{eV/c}^2$.
