In order to understand of halo structures we focused in obtaining a systematic understanding of the quasi-stationary properties of the mass distributions resulting from the gravitational evolution of isolated systems. To this aim we considered controlled numerical experiments in which a system is prepared in a relatively simple initial condition, and it then evolves numerically through gravitational dynamics. In this way we were able to understand several phenomena:

• the mass and energy ejection in cold collapses
• the difference between warm and cold collaspes,
• the origin of the universal properties of QSS from cold collapses,
• the effect of symmetry breaking,
• the generation of trixiality,
• the generation of angular momentum,
• the formation of satellites,
• the effects of discreteness
• the difference between the properties of QSS generated from a top-down and a bottom-up dynamics.

In addition we studied the case in which the initial  systems breaks spherical symmetry and has some angular momentum and  we showed that in such collision-less dynamics, quite generally, there are formed long-lived transient out-of-equilibrium structures with a rich variety of shapes such as spiral arms with or without bars and/or rings, that can have significant observational signatures and consequences.