  We study mimetic $F(R)$ gravity with potential and Lagrange multiplier
constraint. In the context of these theories, we introduce a reconstruction
technique which enables us to realize arbitrary cosmologies, given the Hubble
rate and an arbitrarily chosen $F(R)$ gravity. We exemplify our method by
realizing cosmologies that are in concordance with current observations (Planck
data) and also well known bouncing cosmologies. The attribute of our method is
that the $F(R)$ gravity can be arbitrarily chosen, so we can have the appealing
features of the mimetic approach combined with the known features of some
$F(R)$ gravities, which unify early-time with late-time acceleration. Moreover,
we study the existence and the stability of de Sitter points in the context of
mimetic $F(R)$ gravity. In the case of unstable de Sitter points, it is
demonstrated that graceful exit from inflation occurs. We also study the
Einstein frame counterpart theory of the Jordan frame mimetic $F(R)$ gravity,
we discuss the general properties of the theory and exemplify our analysis by
studying a quite interesting from a phenomenological point of view, model with
two scalar fields. We also calculate the observational indices of the two
scalar field model, by using the two scalar field formalism. Furthermore, we
extensively study the dynamical system that corresponds to the mimetic $F(R)$
gravity, by finding the fixed points and studying their stability. Finally, we
modify our reconstruction method to function in the inverse way and thus
yielding which $F(R)$ gravity can realize a specific cosmological evolution,
given the mimetic potential and the Lagrange multiplier.
