Non-baryonic Dark Matter (DM) appears in galaxies and other cosmic structures when and only when the acceleration of gravity, as computed considering only baryons, goes below a well defined value a0 = 1.2 × 10-8 cm s-2. This fact is extremely important and suggestive of the possibility of a breakdown of Newton's law of gravity (or inertia) below a0. It is therefore important to verify whether Newton's law of gravity holds in this regime of accelerations. In order to do this, one has to study the dynamics of objects that do not contain significant amounts of DM and therefore should follow Newton's prediction for whatever small accelerations. Globular clusters are believed, even by strong supporters of DM, to contain negligible amounts of DM and therefore are ideal for testing Newtonian dynamics in the low acceleration limit. Here, we discuss the status of an ongoing program aimed to do this test. Compared to other studies of globular clsuters, the novelty is that we trace the velocity dispersion profile of globular clusters far enough from the center to probe gravitational accelerations well below a0. In all three clusters studied so far the velocity dispersion is found to remain constant at large radii rather than follow the Keplerian falloff. On average, the flattening occurs at the radius where the cluster internal acceleration of gravity is 1.8 +/- 0.4 × 10-8 cm s-2, fully consistent with MOND predictions.