Fundamental models of various physical media based on configurations with two parallel cores (waveguides) amount to systems of linearly coupled equations. A known example is the model of dual-core optical fibers or planar waveguides, which amounts to a system of coupled nonlinear Schroedinger (NLS) equations. In such systems, obvious symmetric soliton solutions lose their stability through symmetry-breaking bifurcations, when the energy exceeds a certain critical value. The bifurcations simultaneously gives rise to stable asymmetric solitons. The talk aims to present an overview of basic models, results, and physical applications of the symmetry-breaking phenomena in conservative and dissipative nonlinear media.