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Binary interactions lead to the formation of intriguing objects, such as compact binaries, super-novae, gamma ray bursts, X-ray binaries, pulsars, novae, cataclysmic variables, hot subdwarf stars, barium stars and blue stragglers. To study the evolution of binary populations and the consequent formation of these objects, many methods have been developed over the years, for which a robust approach named binary pop-ulation synthesis (BPS) warrants special attention. This approach has seen widespread application in many areas of astrophysics, including but not limited to analyses of the stellar content of galaxies, research on galactic chemical evolution and studies concerning star formation and cosmic re-ionization. In this review,we discuss the role of BPS, its general picture and the various components that comprise it. We pay special attention to the stability criteria for mass transfer in binaries, as this stability largely determines the fate of binary systems. We conclude with our perspectives regarding the future of this field.