Trapped ultracold atomic gases are an ideal system for the study of nonlinear effects and non-equilibrium physics in an interacting many-body quantum system. Generally, such systems consist of a Bose-Einstein condensate co-existing with a thermal atomic cloud, with the interactions and dynamics between these two subsystems creating interesting dynamical situations, broadly applicable to a number of physical systems (e.g. ultracold atoms, exciton-polariton condensates, liquid Helium). This talk focuses on selected experimentally-relevant aspects of non-equilibrium physics in such systems. Specifically, I will discuss (i) the structure and dynamics of macroscopic excitations (solitons, vortices) in such systems, (ii) the emergence of universal scaling laws, associated with spontaneous defect creation following a quench across the phase transition, and (iii) the emergence of phase separation and pattern formation for the case of a two-component quantum gas.