Tournaments are popularly used in sporting events to select a champion. They are also used in experiments where paired comparison procedures are needed. Knockout tournaments are most useful when the number of treatments is too large to use the more well-known round-robin tournament. The mathematical and statistical literature does not address questions such as how one should seed teams in a tournament or which tournament structures are appropriate. For double-elimination (DE) tournaments even basic questions such as how many structures there are remain unanswered. This article addresses some fundamental questions concerning DE tournaments, including the number of DE tournaments and the probabilities of teams winning a DE tournament. Edwards gives many results about single-elimination (SE) tournaments, such as the probability of winning an SE tournament and a notation for labeling and counting them. In this article I develop similar results for the DE tournaments. The results given apply to an arbitrary number of teams, and not just four or eight, as is popular in the literature.