  We present a general formulation of the theory for a non-minimally coupled
perfect fluid in which both conformal and disformal couplings are present. We
discuss how such non-minimal coupling is compatible with the assumptions of a
perfect fluid and derive both the Einstein and the fluid equations for such
model. We found that, while the Euler equation is significantly modified with
the introduction of an extra force related to the local gradients of the
curvature, the continuity equation is unaltered, thus allowing for the
definition of conserved quantities along the fluid flow. As an application to
cosmology and astrophysics we compute the effects of the non-minimal coupling
on a Friedmann--Lema\^itre--Robertson--Walker metric at both background and
linear perturbation level and on the Newtonian limit of our theory.
