  We investigate the possibility that Dark Matter arises as a composite state
of a fundamental confining dynamics, together with the Higgs boson. We focus on
the minimal SU(4)$\times$SU(4)/SU(4) model which has both a Dark Matter and a
Higgs candidates arising as pseudo-Nambu-Goldstone bosons. At the same time, a
simple underlying gauge-fermion theory can be defined providing an existence
proof of, and useful constraints on, the effective field theory description. We
focus on the parameter space where the Dark Matter candidate is mostly a gauge
singlet. We present a complete calculation of its relic abundance and find
preferred masses between 500 GeV to a few TeV. Direct Dark Matter detection
already probes part of the parameter space, ruling out masses above 1 TeV,
while Indirect Detection is relevant only if non-thermal production is assumed.
The prospects for detection of the odd composite scalars at the LHC are also
established.
