The report is about perturbations of the equations of motion of the teleparallel Hayashi-Shirafuji theory of gravity, which was first proposed in 1979 as a generalization of General Relativity. First, the speaker will demonstrate from what principles the action of this theory is derived. Then he will illustrate that there is a spherically symmetric solution in this theory, that reproduces the universal gravitational law. The main part of the report is devoted to the results associated with perturbations of the equations of motion over a flat Minkowski background and cosmological perturbations of Friedmann space in the case when matter is represented by an ideal liquid. Perturbative analysis of the equations of motion allows us to identify those realizations of the theory that can lay claim to be an adequate generalization of Einstein’s theory of gravity.