  We undertook coordinated campaigns with the Green Bank, Effelsberg, and
Arecibo radio telescopes during Chandra X-ray Observatory and XMM-Newton
observations of the repeating fast radio burst FRB 121102 to search for
simultaneous radio and X-ray bursts. We find 12 radio bursts from FRB 121102
during 70 ks total of X-ray observations. We detect no X-ray photons at the
times of radio bursts from FRB 121102 and further detect no X-ray bursts above
the measured background at any time. We place a 5$\sigma$ upper limit of
$3\times10^{-11}$ erg cm$^{-2}$ on the 0.5--10 keV fluence for X-ray bursts at
the time of radio bursts for durations $<700$ ms, which corresponds to a burst
energy of $4\times10^{45}$ erg at the measured distance of FRB 121102. We also
place limits on the 0.5--10 keV fluence of $5\times10^{-10}$ erg cm$^{-2}$ and
$1\times10^{-9}$ erg cm$^{-2}$ for bursts emitted at any time during the
XMM-Newton and Chandra observations, respectively, assuming a typical X-ray
burst duration of 5 ms. We analyze data from the Fermi Gamma-ray Space
Telescope Gamma-ray Burst Monitor and place a 5$\sigma$ upper limit on the
10--100 keV fluence of $4\times10^{-9}$ erg cm$^{-2}$ ($5\times10^{47}$ erg at
the distance of FRB 121102) for gamma-ray bursts at the time of radio bursts.
We also present a deep search for a persistent X-ray source using all of the
X-ray observations taken to date and place a 5$\sigma$ upper limit on the
0.5--10 keV flux of $4\times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ ($3\times10^{41}$
erg~s$^{-1}$ at the distance of FRB 121102). We discuss these non-detections in
the context of the host environment of FRB 121102 and of possible sources of
fast radio bursts in general.

High Energy Astrophysical Phenomena