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These lectures address the dynamics of phase ordering out of equilibrium incondensed matter and in quantum field theory in cosmological settings,emphasizing their similarities and differences. In condensed matter we describethe phenomenological approach based on the Time Dependent Ginzburg-Landau(TDGL) description. After a general discussion of the main experimental andtheoretical features of phase ordering kinetics and the description of linear(spinodal) instabilities we introduce the scaling hypothesis and show how adynamical correlation length emerges in the large N limit in condensed mattersystems. The large N approximation is a powerful tool in quantum field theorythat allows the study of non-perturbative phenomena in a consistent manner. Westudy the exact solution to the dynamics after a quench in this limit inMinkowski space time and in radiation dominated Friedman-Robertson-WalkerCosmology. There are some remarkable similarities between these very differentsettings such as the emergence of a scaling regime and of a dynamicalcorrelation length at late times that describe the formation and growth ofordered regions. In quantum field theory and cosmology this length scale isconstrained by causality and its growth in time is also associated withcoarsening and the onset of a condensate. We provide a density matrixinterpretation of the formation of defects and the classicalization of quantumfluctuations.

Non-Equilibrium Phase Transitions in Condensed Matter and Cosmology: Spinodal Decomposition, Condensates and Defects