  Theories of massive gravity inevitably include an auxiliary reference metric.
Generically, they also contain an inconsistency known as the Boulware-Deser
ghost. Recently, a family of non-linear massive gravity actions, formulated
with a flat reference metric, were proposed and shown to be ghost free at the
complete non-linear level. In this paper we consider these non-linear massive
gravity actions but now formulated with a general reference metric. We extend
the proof of the absence of the Boulware-Deser ghost to this case. The analysis
is carried out in the ADM formalism at the complete non-linear level. We show
that in these models there always exists a Hamiltonian constraint which, with
an associated secondary constraint, eliminates the ghost. This result
considerably extends the range of known consistent non-linear massive gravity
theories. In addition, these theories can also be used to describe a massive
spin-2 field in an arbitrary, fixed gravitational background. We also discuss
the positivity of the Hamiltonian.
