A compelling solution to the Dark Matter question is that DM particles are part of a “dark sector” with fields uncharged under the Standard Model (SM) gauge group. On the other hand, conformal field theories are ubiquitous and appear as fixed points of gauge theories. CFTs are scale-invariant and UV complete. I will discuss our work on dark sectors that are described by conformal field theories (which cannot generically be described by the particle framework) and couple to the SM via portal interactions. We make generic, model-independent predictions for the nature of dark matter, and show that light DM states are populated in the early universe via freeze-in in such models. The mass gap generated is directly proportional to the coupling to the SM, which makes the model quite minimal. If time allows, I will also briefly comment on another `non-particle' dark matter framework: the gapped continuum.