The role of gravity is crucial in astrophysics. It determines the evolution of any system, over an enormous range of time and space scales. Astronomical stellar systems as composed by N interacting bodies represent examples of self-gravitating systems, usually treatable with the aid of newtonian gravity but for particular cases. In this note I will briefly discuss some of the open problems in the dynamical study of classic self-gravitating N-body systems, over the astronomical range of N. I will also point out how modern research in this field compulsorily requires a heavy use of large scale computations, due to the contemporary requirement of high precision and high computational speed.