  We discuss ghost free models of the recently suggested mimetic dark matter
theory. This theory is shown to be a conformal extension of Einstein general
relativity. Dark matter originates from gauging out its local Weyl invariance
as an extra degree of freedom which describes a potential flow of the
pressureless perfect fluid. For a positive energy density of this fluid the
theory is free of ghost instabilities, which gives strong preference to stable
configurations with a positive scalar curvature and trace of the matter stress
tensor. Instabilities caused by caustics of the geodesic flow, inherent in this
model, serve as a motivation for an alternative conformal extension of Einstein
theory, based on the generalized Proca vector field. A potential part of this
field modifies the inflationary stage in cosmology, whereas its rotational part
at the post inflationary epoch might simulate rotating flows of dark matter.
