We initiate the study of dark matter (DM) models based on a gapped continuum. Dark matter consists of a mixture of states with a continuous mass distribution, which evolves as the universe expands. We present an effective field theory describing the gapped continuum, outline the structure of the Hilbert space and show how to deal with the thermodynamics of such a system. This formalism enables us to study the cosmological evolution and phenomenology of gapped continuum DM in detail. As a concrete example, we consider a weakly interacting continuum (WIC) model, a gapped continuum counterpart of the familiar weakly interacting massive particle. The DM interacts with the Standard Model via a Z portal. The model successfully reproduces the observed relic density, while direct detection constraints are avoided due to the effect of continuum kinematics. The model has striking observational consequences, including continuous decays of DM states throughout cosmological history, as well as cascade decays of DM states produced at colliders. We also describe how the WIC theory can arise from a local, unitary scalar quantum field theory propagating on a five-dimensional warped background with a soft wall.