In this review article, we illustrate the main idea of many-body conformal dynamics and two surprises. Conformal dynamics are non-equilibrium phenomena occurring when inter particle interactions are scale invariant and the particular many-body systems we will focus on further exhibit a non-relativistic Galilean invariance. The first surprise is that in conformal expansion dynamics driven by unitary evolutions of an isolated quantum system, both the energy and entropy can be conserved as opposed to what happen in more familiar free expansions and adiabatic expansions in thermodynamics. Such dynamics in strongly interacting quantum systems where interactions are scale symmetric appear as a very rare example of reversible far-away-from equilibrium many-body dynamics. The second surprise is that conformal dynamics can also emerge in many strongly interacting systems in the large time limit as an asymptotic dynamic symmetry, even though scale symmetries are explicitly broken by strong interactions. The emergent infrared conformal dynamics, purely induced by real-time dynamic renormalization effects, can occur in both low and high dimensions but rely crucially on the scaling dimensions of interaction operators and infrared stability of the scale invariant fixed points.