  We build models where Dark Matter candidates arise as composite states of a
new confining gauge force, stable thanks to accidental symmetries. Restricting
to renormalizable theories compatible with SU(5) unification, we find 13 models
based on SU(N) gauge theories and 9 based on SO(N). We also describe other
models that require non-renormalizable interactions. The two gauge groups lead
to distinctive phenomenologies: SU(N) theories give complex DM, with
potentially observable electric and magnetic dipole moments that lead to
peculiar spin-independent cross sections; SO(N) theories give real DM, with
challenging spin-dependent cross sections or inelastic scatterings. Models with
Yukawa couplings also give rise to spin-independent direct detection mediated
by the Higgs boson and to electric dipole moments for the electron. In some
models DM has higher spin. Each model predicts a specific set of lighter
composite scalars, possibly observable at colliders.
