A collection of author's papers on low-energy quantum gravity is presented in the book. These papers were written in the last decade. In the papers, main results of author's work in a non-geometrical approach to quantum gravity are described, among them: the quantum mechanism of classical gravity giving a possibility to compute the Newton constant; asymptotic freedom at short distances; an interaction of photons with the graviton background leading to the important cosmological consequences; the time delay of photons due to interactions with gravitons. A quantum mechanism of classical gravity based on an existence of this sea of gravitons is described for the Newtonian limit. This mechanism needs graviton pairing and "an atomic structure" of matter for working it. If the considered quantum mechanism of classical gravity is realized in the nature, then an existence of black holes contradicts to Einstein's equivalence principle. It is shown that in this approach the two fundamental constants - Hubble's and Newton's ones - should be connected between themselves. The theoretical values of the constants are computed. In this approach, every massive body would be decelerated due to collisions with gravitons that may be connected with the Pioneer anomaly and the problem of dark matter. It is shown that the predicted and observed values of deceleration of NASA probes are in good agreement. It is also shown by the author that if gravitons are super-strong interacting particles and the low-temperature graviton background exists, the basic present-day cosmological conjecture about the nature of redshifts may be false. In this case, a full magnitude of cosmological redshift would be caused by interactions of photons with gravitons. Non-forehead collisions with gravitons will lead to a very specific additional relaxation of any photonic flux. It gives a possibility of another interpretation of supernovae 1a data - without any kinematics and introduction of dark energy. A few possibilities to verify model's predictions are considered: the specialized ground-based laser experiment; a deceleration of massive bodies and the Pioneer anomaly; a non-universal character of the Hubble diagram for soft and hard radiations; galaxy/quasar number counts.