The study of Voting Systems and Collective Choice lies at the crossroad of social and exact sciences. The former ones focus on the development of electoral systems in the context of the changing needs of societies; the latter ones deal with the formal study of electoral mechanisms, whose underlying axioms reflect universal principles such as equity, representation, stability, and consistency. Different quantitative approaches to the study of electoral systems have been developed: game-theoretic, decision-theoretic, statistical, probabilistic, combinatorial, geometric, and optimization ones. All the authors are prominent scholars from these disciplines. Quantitative approaches offer a powerful tool to detect inconsistencies or poor performance in actual systems. Applications to concrete settings such as EU, American Congress, regional, and committee voting are discussed.