  Fast radio bursts (FRBs) are short (millisecond) radio pulses originating
from enigmatic sources at extragalactic distances so far lacking a detection in
other energy bands. Magnetized neutron stars (magnetars) have been considered
as the sources powering the FRBs, but the connection is controversial because
of differing energetics and the lack of radio and X-ray detections with similar
characteristics in the two classes. We report here the detection by the AGILE
satellite on April 28, 2020 of an X-ray burst in coincidence with the very
bright radio burst from the Galactic magnetar SGR 1935+2154. The burst detected
by AGILE in the hard X-ray band (18-60 keV) lasts about 0.5 seconds, it is
spectrally cutoff above 80 keV, and implies an isotropically emitted energy ~
$10^{40}$ erg. This event is remarkable in many ways: it shows for the first
time that a magnetar can produce X-ray bursts in coincidence with FRB-like
radio bursts; it also suggests that FRBs associated with magnetars may emit
X-ray bursts of both magnetospheric and radio-pulse types that may be
discovered in nearby sources. Guided by this detection, we discuss SGR
1935+2154 in the context of FRBs, and especially focus on the class of
repeating-FRBs. Based on energetics, magnetars with fields B ~ $10^{15}$ G may
power the majority of repeating-FRBs. Nearby repeating-FRBs offer a unique
occasion to consolidate the FRB-magnetar connection, and we present new data on
the X-ray monitoring of nearby FRBs. Our detection enlightens and constrains
the physical process leading to FRBs: contrary to previous expectations,
high-brightness temperature radio emission coexists with spectrally-cutoff
X-ray radiation.
