  Observations at Very High Energies (VHE, E > 100 GeV) of the BL Lac object
Mrk 501 taken with the High Energy Stereoscopic System (H.E.S.S.) in four
distinct periods between 2004 and 2014 are presented, with focus on the 2012
and 2014 flaring states. The source is detected with high significance above
$\sim$ 2 TeV in $\sim$ 13.1 h livetime. The observations comprise low flux
states and strong flaring events, which in 2014 show a flux level comparable to
the 1997 historical maximum. Such high flux states enable spectral variability
and flux variability studies down to a timescale of four minutes in the 2-20
TeV energy range. During the 2014 flare, the source is clearly detected in each
of these bins. The intrinsic spectrum is well described by a power law of index
$\Gamma=2.15\pm0.06$ and does not show curvature in this energy range. Flux
dependent spectral analyses show a clear harder-when-brighter behaviour. The
high flux levels and the high sensitivity of H.E.S.S. allow studies in the
unprecedented combination of short timescales and an energy coverage that
extends significantly above 20 TeV. The high energies allow us to probe the
effect of EBL absorption at low redshifts, jet physics and LIV. The
multiwavelength context of these VHE observations is presented as well.
