In order to give a precise and general formulation to the strong equivalence principle (SEP) the authors define, in an appropriate inertial frame and in the slow-motion approximation, a local gravitational system. They say that the SEP is fulfilled if, when the size r of the system is sufficiently small, its dynamical behaviour, to a given accuracy, is universal and not affected by the external world. In the theory of general relativity, along with the Newtonian tidal force Ftid varies as r, there exists a non-linear, relativistic contribution Fnl which, in order of magnitude, is independent of r; this seems to leave a trace of the external world in an arbitrarily small gravitating system and threatens to violate the principle. It is shown, however, that Fnl is always smaller than Ftid, at least in the weak-field slow-motion approximation and hence, in this approximation, the SEP survives. In other metric theories of gravity, however, violations of the SEP occur. A notion of universal gravitational clocks is briefly discussed.