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[en] 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 5σ upper limit of 3 × 10−11 erg cm−2 on the 0.5–10 keV fluence for X-ray bursts at the time of radio bursts for durations ms, which corresponds to a burst energy of 4 × 1045 erg at the measured distance of FRB 121102. We also place limits on the 0.5–10 keV fluence of 5 × 10−10 and 1 × 10−9 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 5σ upper limit on the 10–100 keV fluence of 4 × 10−9 erg cm−2 (5 × 1047 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 5σ upper limit on the 0.5–10 keV flux of 4 × 10−15 erg s−1 cm−2 (3 × 1041 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.