  Bimetric gravity, is a theory of gravity that posits the existence of two
interacting and dynamical metric tensors. The spectrum of bimetric gravity
consists of a massless and a massive spin-2 particle. The form of the
interactions between the two metrics $g_{\mu\nu}$ and $f_{\mu\nu}$ is
constrained by requiring absence of the so called Boulware-Deser ghost. In this
work we extend the original bimetric theory to its bimetric-affine counterpart,
in which the two connections, associated with the Ricci scalars, are treated as
independent variables. We examine in detail the case of an additional quadratic
in the Ricci scalar curvature term $\mathcal{R}^2(g,\Gamma)$ and we find that
this theory is free of ghosts for a wide range of the interaction parameters,
not excluding the possibility of a Dark Matter interpretation of the massive
spin-2 particle.
