  Theoretical considerations of fundamental physics, as well as certain
cosmological observations, persistently point out to permissibility, and maybe
necessity, of macroscopic modifications of the Einstein general relativity. The
field-theoretical formulation of general relativity helped us to identify the
phenomenological seeds of such modifications. They take place in the form of
very specific mass-terms, which appear in addition to the field-theoretical
analog of the usual Hilbert-Einstein Lagrangian. We interpret the added terms
as masses of the spin-2 and spin-0 gravitons. The arising finite-range gravity
is a fully consistent theory, which smoothly approaches general relativity in
the massless limit, that is, when both masses tend to zero and the range of
gravity tends to infinity. We show that all local weak-field predictions of the
theory are in perfect agreement with the available experimental data. However,
some other conclusions of the non-linear massive theory are in a striking
contrast with those of general relativity. We show in detail how the
arbitrarily small mass-terms eliminate the black hole event horizon and replace
a permanent power-law expansion of a homogeneous isotropic universe with an
oscillatory behaviour. One variant of the theory allows the cosmological scale
factor to exhibit an `accelerated expansion'instead of slowing down to a
regular maximum of expansion. We show in detail why the traditional,
Fierz-Pauli, massive gravity is in conflict not only with the static-field
experiments but also with the available indirect gravitational-wave
observations. At the same time, we demonstrate the incorrectness of the widely
held belief that the non-Fierz-Pauli theories possess `negative energies' and
`instabilities'.
