The problem of the existence and the qualitative properties of parabolic and other selected trajectories (periodic, collision) for the N-body problem (from the classical celestial mechanics point of view to more recent advances in molecular and quantum models) has been extensively studied over the decades, and, more recently, new tools and approaches have given a significant boost to the field. We shall review some old an new results on the existence and classication of selected trajectories of the classical N-centre and N-body problem, with an emphasis on new analytical and geometrical techniques.