Universality near continuous phase transitions has led to a tremendous increase in our understanding of the fundamental aspects governing the large scale structure formation of seemingly different systems. This insensitivity to microscopic details culminated in the renormalization group (RG) program of quantum field theory, enabling a comprehensive classification of systems based on their universal properties near fixed points of the RG flow. We discuss implications of universal aspects on the real time evolution during and after rapidly driving a system through a phase transition. In particular, we present results on (i) defect nucleation and its connection to the Kibble-Zurek mechanism for rapidly cooling quenched one-dimensional Bose gases and coupled quantum wires and (ii) universal self-similar scaling dynamics, e.g. associated with the approach of a non-thermal fixed point, during the relaxation of isolated or open systems driven far from equilibrium.