  The Gamma Ray Burst (GRB) 180720B is one of the brightest events detected by
the Fermi satellite and the first GRB detected by the H.E.S.S. telescope above
100 GeV. We analyse the Fermi (GBM and LAT) and Swift (XRT and BAT) data and
describe the evolution of the burst spectral energy distribution in the 0.5 keV
- 10 GeV energy range over the first 500 seconds of emission. We reveal a
smooth transition from the prompt phase, dominated by synchrotron emission in a
moderately fast cooling regime, to the afterglow phase whose emission has been
observed from the radio to the GeV energy range. The LAT (0.1 - 100 GeV) light
curve initially rises ($F_{\rm LAT}\propto t^{2.4}$), peaks at $\sim$78 s, and
falls steeply ($F_{\rm LAT}\propto t^{-2.2}$) afterwards. The peak, which we
interpret as the onset of the fireball deceleration, allows us to estimate the
bulk Lorentz factor $\Gamma_{0}\sim 150 \ (300)$ under the assumption of a
wind-like (homogeneous) circum-burst medium density. We derive a flux upper
limit in the LAT energy range at the time of H.E.S.S. detection, but this does
not allow us to unveil the nature of the high energy component observed by
H.E.S.S. We fit the prompt spectrum with a physical model of synchrotron
emission from a non-thermal population of electrons. The 0 - 35 s spectrum
after its $E F(E)$ peak (at 1 - 2 MeV) is a steep power law extending to
hundreds of MeV. We derive a steep slope of the injected electron energy
distribution $N(\gamma)\propto \gamma^{-5}$. Our fit parameters point towards a
very low magnetic field ($B'\sim 1 $ G) in the emission region.
