With $\mathcal{H}$ a family of sets and $\mathcal{D}$ a set, a social choice function assigns a nonempty subset $F(Y,D)$ of Y to each $(Y,D) \in \mathcal{H} \times \mathcal{D}$. In social choice theory, the sets in $\mathcal{H}$ are feasible alternative sets, each $D \in \mathcal{D}$ specifies preferences on the alternatives of individuals in the society, and $F(Y,D)$ denotes the “best” alternatives in Y for each potential situation $(Y,D)$. This paper reviews a number of topics in social choice theory through the medium of social choice functions. It begins with two-alternative situations, providing axiomatic characterizations of simple majority, weighted majority, absolute special majority and representative systems. Various aspects of simple majorities with many alternatives are examined, including facets of cyclical majorities, single-peaked preferences and sets of individual preference orders that guarantee the transitivity of simple majority. A classification of types of conditions for general socia...