  We consider theories in which a dark sector is described by a Conformal Field
Theory (CFT) over a broad range of energy scales. A coupling of the dark sector
to the Standard Model breaks conformal invariance. While weak at high energies,
the breaking grows in the infrared, and at a certain energy scale the theory
enters a confined (hadronic) phase. One of the hadronic excitations can play
the role of dark matter. We study a "Conformal Freeze-In" cosmological
scenario, in which the dark sector is populated through its interactions with
the SM at temperatures when it is conformal. In this scenario, the dark matter
relic density is determined by the CFT data, such as the dimension of the CFT
operator coupled to the Standard Model. We show that this simple and highly
predictive model of dark matter is phenomenologically viable. The observed
relic density is reproduced for a variety of SM operators ("portals") coupled
to the CFT, and the resulting models are consistent with observational
constraints. The mass of the COFI dark matter candidate is predicted to be in
the keV-MeV range.
