The Marquis de Condorcet, a French philosopher, mathematician, and political scientist, studied mathematical aspects of voting in the eighteenth century. Condorcet was interested in studying voting rules as procedures for aggregating noisy signals and in the paradoxical nature of ranking three or more alternatives. We survey some of the main mathematical models, tools, and results in a theory that studies probabilistic aspects of social choice. Our journey will take us through major results in mathematical economics from the second half of the twentieth century, through the theory of Boolean functions and their influences and through recent results in Gaussian geometry and functional inequalities.