  We perform a systematic study of various versions of massive gravity with and
without violation of Lorentz symmetry in arbitrary dimension. These theories
are well known to possess very unusual properties, unfamiliar from studies of
gauge and Lorentz invariant models. These peculiarities are caused by mixing of
familiar transverse fields with revived longitudinal and pure gauge
(Stueckelberg) fields and are all seen already in quadratic approximation. They
are all associated with non-trivial dispersion laws, which easily allow
superluminal propagation, ghosts, tachyons and essential irrationalities.
Moreover, coefficients in front of emerging modes are small, what makes the
theories essentially non-perturbative within a large Vainshtein radius.
Attempts to get rid of unwanted degrees of freedom by giving them infinite
masses lead to DVZ discontinuities in parameter (moduli) space, caused by
un-permutability of different limits. Also, the condition m_{gh}=\infty can not
be preserved already in non-trivial gravitational backgrounds and is unstable
under any other perturbations of linearized gravity. At the same time an {\it a
priori} healthy model of massive gravity in quadratic approximation definitely
exists: provided by any mass level of Kaluza-Klein tower. It bypasses the
problems because gravity field is mixed with other fields, and this explains
why such mixing helps in other models. At the same time this can imply that the
really healthy massive gravity can still require infinite number of extra
fields beyond quadratic approximation.
