Spontaneous symmetry breaking is the foundation of electroweak unification and serves as an integral part of the model building beyond the standard model of particle physics and it also finds interesting applications in the late Universe. We review development related to obtaining the late cosmic acceleration from spontaneous symmetry breaking in the Universe at large scales. This phenomenon is best understood through Ginzburg-Landau theory of phase transitions which we briefly describe. Hereafter, we present elements of spontaneous symmetry breaking in relativistic field theory. We then discuss the "symmetron"scenario-based upon symmetry breaking in the late Universe which is realized by using a specific form of conformal coupling. However, the model is faced with "NO GO"for late-time acceleration due to local gravity constraints. We argue that the problem can be circumvented by using the massless λφ4 theory coupled to massive neutrino matter. As for the early Universe, spontaneous symmetry breaking finds its interesting applications in the study of electroweak phase transition. To this effect, we first discuss in detail the Ginzburg-Landau theory of first-order phase transitions and then apply it to electroweak phase transition including technical discussions on bubble nucleation and sphaleron transitions. We provide a pedagogical exposition of dynamics of electroweak phase transition and emphasize the need to go beyond the standard model of particle physics for addressing the baryogenesis problem. Review ends with a brief discussion on Affleck-Dine mechanism and spontaneous baryogenesis. Appendixes include technical details on essential ingredients of baryogenesis, sphaleron solution, one-loop finite temperature effective potential and dynamics of bubble nucleation.