Scaling ideas and renormalization group approaches proved crucial for a deep understanding and classification of critical phenomena in thermal equilibrium. Over the past decades, these powerful conceptual and mathematical tools were extended to continuous phase transitions separating distinct non-equilibrium stationary states in driven classical and quantum systems. In concordance with detailed numerical simulations and laboratory experiments, several prominent dynamical universality classes have emerged that govern large-scale, long-time scaling properties both near and far from thermal equilibrium. These pertain to genuine specific critical points as well as entire parameter space regions for steady states that display generic scale invariance. The exploration of non-stationary relaxation properties and associated physical aging scaling constitutes a complementary potent means to characterize cooperative dynamics in complex out-of-equilibrium systems. This article describes dynamic scaling features through paradigmatic examples that include near-equilibrium critical dynamics, driven lattice gases and growing interfaces, correlation-dominated reaction-diffusion systems, and basic epidemic models.