The theory of phase ordering kinetics, or 'domain coarsening', describes the growth of order in systems quenched from a homogeneous phase into a broken-symmetry phase. This process is governed by the competition between different broken symmetry phases to reach equilibrium. The focus is on the scaling regime that develops at long times after the quench, where the growth laws of characteristic length scales and the form of scaling functions are determined. The review emphasizes systems with complex order parameters, such as vector and tensor fields, which are essential for describing phenomena like phase ordering in nematic liquid crystals. Topological defects, including domain walls, vortices, and strings, provide a unifying framework for understanding coarsening in various systems. The review discusses dynamical models, including the Allen-Cahn equation for non-conserved fields and the Cahn-Hilliard equation for conserved fields. It explores the scaling hypothesis, which posits that at long times, the domain structure is statistically independent of time when scaled by a characteristic length scale L(t). The structure factor and pair correlation function are analyzed to determine the scaling forms of these functions. The growth laws for L(t) are derived using the scaling hypothesis and the tail of the structure factor, leading to power-law scaling for both conserved and non-conserved fields. The Lifshitz-Slyozov-Wagner theory is discussed, showing that the characteristic size of minority phase droplets increases like t^(1/3) in the limit of small volume fraction. The theory also considers the dynamics of droplet size distributions, leading to a scaling function for the droplet size distribution. The review also addresses the extension of these results to binary liquids, where advection by fluid flow is important. The analysis includes the effects of hydrodynamic flow on the order parameter and the resulting equations of motion for the system. The review concludes with a discussion of the universality classes and the role of topological defects in phase ordering dynamics.The theory of phase ordering kinetics, or 'domain coarsening', describes the growth of order in systems quenched from a homogeneous phase into a broken-symmetry phase. This process is governed by the competition between different broken symmetry phases to reach equilibrium. The focus is on the scaling regime that develops at long times after the quench, where the growth laws of characteristic length scales and the form of scaling functions are determined. The review emphasizes systems with complex order parameters, such as vector and tensor fields, which are essential for describing phenomena like phase ordering in nematic liquid crystals. Topological defects, including domain walls, vortices, and strings, provide a unifying framework for understanding coarsening in various systems. The review discusses dynamical models, including the Allen-Cahn equation for non-conserved fields and the Cahn-Hilliard equation for conserved fields. It explores the scaling hypothesis, which posits that at long times, the domain structure is statistically independent of time when scaled by a characteristic length scale L(t). The structure factor and pair correlation function are analyzed to determine the scaling forms of these functions. The growth laws for L(t) are derived using the scaling hypothesis and the tail of the structure factor, leading to power-law scaling for both conserved and non-conserved fields. The Lifshitz-Slyozov-Wagner theory is discussed, showing that the characteristic size of minority phase droplets increases like t^(1/3) in the limit of small volume fraction. The theory also considers the dynamics of droplet size distributions, leading to a scaling function for the droplet size distribution. The review also addresses the extension of these results to binary liquids, where advection by fluid flow is important. The analysis includes the effects of hydrodynamic flow on the order parameter and the resulting equations of motion for the system. The review concludes with a discussion of the universality classes and the role of topological defects in phase ordering dynamics.