Globular clusters are collisional systems, where stars of different masses orbit and mutually interact. They are the best “natural laboratories” in the Universe for studying multi-body dynamics and their (reciprocal) effects on stellar evolution. Although these objects have been studied since the very beginning of modern astrophysics, little is known observationally about their internal kinematics, thus preventing a complete understanding of their dynamical state, and of their formation and evolutionary history. We present the first results from the Very Large Telescope (VLT) Multi- Instrument Kinematic Survey of Galactic globular clusters (MIKiS), which is specifically designed to provide line-of-sight velocities of hundreds of individual stars over the entire radial extension of a selected sample of clusters. The survey allows the first kinematical exploration of the innermost regions of high-density globular clusters. When combined with proper motion measurements, it will provide the full 3D view in velocity-space for each system. Long- running open issues, such as the accurate shapes of the velocity dispersion profiles, the existence of systemic rotation and orbital anisotropy (and thus the level of relaxation), and the controversial presence of intermediate-mass black holes in star clusters can finally be addressed, impacting our understanding of the formation and evolutionary processes of globular clusters and their interactions with the Galactic tidal field.