Conventionally, the main focus for the cosmic evolution of our universe has been on descriptions in terms of particles: dark matter (DM) as massive particle, and dark radiation, if existing at all, in the form of massless or very light particle. In this talk, I will discuss a scenario where conformal field theory (CFT) plays a crucial role in cosmology, especially production of naturally light dark matter. When the Standard Model (SM) couples to a sector of CFT, I will argue that in the regime where only the SM sector gets reheated, the energy density of the CFT sector is populated via energy injection from the SM (conformal freeze-in) and can be reliably computed. Such CFT energy density, redshifting like a radiation, will not account for DM unless somehow a mass gap is generated in the CFT. Remarkably, such a gap scale is dynamically generated due to phase transitions of SM: electroweak and QCD. In this way, DM energy today is produced in the early universe when it is not in gapped particle state, but a CFT state. Moreover, DM mass scale consistent with observed relic density and other observational constraints (mostly bullet cluster and warm dark matter bounds) turns out to be less than MeV. Therefore, conformal freeze-in provides a framework for naturally light DM.