  We find the Holographic Renormalization Group equations for the holographic
duals of generic gravity theories coupled to form fields and spin-1/2 fermions.
Using Hamilton-Jacobi theory we discuss the structure of Ward identities,
anomalies, and the recursive equations for determining the divergent terms of
the generating functional. In particular, the Ward identity associated to
diffeomorphism invariance contains an anomalous contribution that, however, can
be solved either by a suitable counter term or by imposing a condition on the
boundary fields. Consistency conditions for the existence of the dual arise, if
one requires that a Callan-Symanzik type equation follows from the Hamiltonian
constraint. Under mild assumptions we are able to find a class of solutions to
the constraint equations. The structure of the fermionic phase space and
constraints is treated extensively for any dimension and signature.
