  We present the first extensive radio to gamma-ray observations of a
fast-rising blue optical transient (FBOT), AT2018cow, over its first ~100 days.
AT2018cow rose over a few days to a peak luminosity $L_{pk}\sim4\times 10^{44}$
erg/s exceeding those of superluminous supernovae (SNe), before declining as
$\propto t^{-2}$. Initial spectra at $\lesssim 15$ days were mostly featureless
and indicated large expansion velocities v~0.1c and temperatures reaching 30000
K. Later spectra revealed a persistent optically-thick photosphere and the
emergence of H and He emission features with v~sim 4000 km/s with no evidence
for ejecta cooling. Our broad-band monitoring revealed a hard X-ray spectral
component at $E\ge 10$ keV, in addition to luminous and highly variable soft
X-rays, with properties unprecedented among astronomical transients. An abrupt
change in the X-ray decay rate and variability appears to accompany the change
in optical spectral properties. AT2018cow showed bright radio emission
consistent with the interaction of a blastwave with $v_{sh}$~0.1c with a dense
environment ($\dot M\sim10^{-3}-10^{-4}\,M_{\odot}yr^{-1}$ for $v_w=1000$
km\s). While these properties exclude Ni-powered transients, our
multi-wavelength analysis instead indicates that AT2018cow harbored a "central
engine", either a compact object (magnetar or black hole) or an embedded
internal shock produced by interaction with a compact, dense circumstellar
medium. The engine released $\sim10^{50}-10^{51.5}$ erg over $\sim10^3-10^5$ s
and resides within low-mass fast-moving material with equatorial-polar density
asymmetry ($M_{ej,fast}\lesssim0.3\,\rm{M_{\odot}}$). Successful SNe from
low-mass H-rich stars (like electron-capture SNe) or failed explosions from
blue supergiants satisfy these constraints. Intermediate-mass black-holes are
disfavored by the large environmental density probed by the radio observations.
