In several recent works the symmetries of the n-body equation have been exploited in order to find new periodic solutions using a variational principle with symmetries in time and space.Such solutions are the natural generalization of the relative equilibrium motions,the well known periodic orbits for the classical problem.The variational approach to the search of periodic solutions consists in finding critical points of the associated action or Maupertuis functionals.Though the presence of singularities has to be held responsible of the hardest difficulties in finding critical points,it is also the ultimate cause of their existence.An intriguing aspect of the variational approach to the periodic n-body problem is that it involves the study of a variety of issues: analytical,algebraic,topological and computational.Sharp level estimates on colliding trajectories are possible when central collisions are known and classified(as in the three-body problem),and so in this field the study of central configurations(relative equilibria)plays a key role.Other ideas and results which are basic for this line of research are suitable regularization theorems and the analysis of colliding trajectories by asymptotic or topological methods,classification theorems for the symmetry groups and related local/global variations for colliding trajectories.We shall focus on collision trajectories,regularization.A special attention will be devoted to the study of parabolic trajectories and their variational characterization.