  The Brout-Englert-Higgs mechanism describes the generation of masses of
fundamental particles in the Standard Model (SM). It predicts the existence of
one scalar particle with precisely predicted couplings to fermions and gauge
bosons. Deviations from these predictions, such as the observation of
additional scalar particles, would indicate non-minimal Higgs sectors and
beyond-the-SM physics. We motivate extended scalar sectors based on theoretical
and observational grounds and discuss their possible manifestations at
colliders or gravitational-wave experiments. We explore smoking-gun signatures
of electroweak baryogenesis at the LHC or a future $e^+e^-$ collider, and we
examine their interplay with the observation of a primordial gravitational-wave
background at LISA. We also consider the possibility of an additional Higgs
boson at about 95 GeV motivated by excesses in diphoton and ditau final states
observed by CMS. Finally, we discuss the impact of the CDF measurement of the
$W$-boson mass on the parameter space of scalar extensions of the SM.
