I review recent studies about the gravitational potential and stellar dynamics of M87 in particular, and the dynamics of the stars in the presence of a super-massive central black hole, in general. At large radii, investigations of both the X-ray emitting gas and the velocity distribution of globular clusters indicate the presence of large amounts of non-luminous matter, possibly belonging to the inner parts of the Virgo cluster. At small radii, there is no evidence from the stellar kinematics, at most a hint, for the existence of a central point mass, whereas the gas dynamics reveal the presence of a highly concentrated mass in the centre of M87, possibly a super-massive black hole (BH). Given the existence of such a central mass, the stellar kinematics indicate a strong tangential anisotropy of the stellar motion inside a few arcseconds. The implications of this result for the evolution and formation history of M87 and its central BH are discussed. I also discuss in more general terms the structural changes that a highly concentrated central mass can induce in its parent galaxy.