  This is the third out of five chapters of the final report [1] of the
Workshop on Physics at HL-LHC, and perspectives on HE-LHC [2]. It is devoted to
the study of the potential, in the search for Beyond the Standard Model (BSM)
physics, of the High Luminosity (HL) phase of the LHC, defined as
$3~\mathrm{ab}^{-1}$ of data taken at a centre-of-mass energy of
$14~\mathrm{TeV}$, and of a possible future upgrade, the High Energy (HE) LHC,
defined as $15~\mathrm{ab}^{-1}$ of data at a centre-of-mass energy of
$27~\mathrm{TeV}$. We consider a large variety of new physics models, both in a
simplified model fashion and in a more model-dependent one. A long list of
contributions from the theory and experimental (ATLAS, CMS, LHCb) communities
have been collected and merged together to give a complete, wide, and
consistent view of future prospects for BSM physics at the considered
colliders. On top of the usual standard candles, such as supersymmetric
simplified models and resonances, considered for the evaluation of future
collider potentials, this report contains results on dark matter and dark
sectors, long lived particles, leptoquarks, sterile neutrinos, axion-like
particles, heavy scalars, vector-like quarks, and more. Particular attention is
placed, especially in the study of the HL-LHC prospects, to the detector
upgrades, the assessment of the future systematic uncertainties, and new
experimental techniques. The general conclusion is that the HL-LHC, on top of
allowing to extend the present LHC mass and coupling reach by $20-50\%$ on most
new physics scenarios, will also be able to constrain, and potentially
discover, new physics that is presently unconstrained. Moreover, compared to
the HL-LHC, the reach in most observables will generally more than double at
the HE-LHC, which may represent a good candidate future facility for a final
test of TeV-scale new physics.
