  Unlike its minimal counterpart, the Next to Minimal supersymmetric Standard
Model (NMSSM) allows the possibility that the lightest neutralino could have a
mass as small as $\sim 1 {\rm GeV}$ while still providing a significant
component of relic dark matter (DM). Such a neutralino can provide an invisible
decay mode to the Higgs as well. Further, the observed SM-like Higgs boson
($H_{125}$) could also have an invisible branching fraction as high as $\sim
19\%$. Led by these facts, we first delineate the region of parameter space of
the NMSSM with a light neutralino ($M_{{\tilde{\chi}}_{1}^{0}} < 62.5 {\rm
GeV}$) that yields a thermal neutralino relic density smaller than the measured
relic density of cold dark matter, and is also compatible with constraints from
collider searches, searches for dark matter, and from flavor physics. We then
examine the prospects for probing the NMSSM with a light neutralino via direct
DM detection searches, via invisible Higgs boson width experiments at future
$e^+e^-$ colliders, via searches for a light singlet Higgs boson in $2b2\mu$,
$2b2\tau$ and $2\mu2\tau$ channels and via pair production of winos or doublet
higgsinos at the high luminosity LHC and its proposed energy upgrade. For this
last-mentioned electroweakino search, we perform a detailed analysis to map out
the projected reach in the $3l+{\rm E{\!\!\!/}_T}$ channel, assuming that
chargino decays to $W {\tilde{\chi}}_{1}^{0}$ and the neutralino(s) decay to
$Z$ or $H_{125}$ + ${\tilde{\chi}}_{1}^{0}$. We find that the HL-LHC can
discover SUSY in just part of the parameter space in each of these channels,
which together can probe almost the entire parameter space. The HE-LHC probes
essentially the entire region with higgsinos (winos) lighter than 1 TeV (2 TeV)
independently of how the neutralinos decay, and leads to significantly larger
signal rates.
