The Banzhaf index of power in a voting situation depends on the number of ways in which each voter can effect a "swing" in the outcome. It is comparable--but not actually equivalent--to the better-known Shapley-Shubik index, which depends on the number of alignments or "orders of support" in which each voter is pivotal. This paper investigates some properties of the Banzhaf index, the main topics being its derivation from axioms and its behavior in weighted-voting models when the number of small voters tends to infinity. These matters have previously been studied from the Shapley-Shubik viewpoint, but the present work reveals some striking differences between the two indexes. The paper also attempts to promote better communication and less duplication of mathematical effort in this field by identifying and describing several other theories, formally equivalent to Banzhaf's, that are found in fields ranging from sociology to electrical engineering. An extensive bibliography is provided.