A tournament on a graph is an orientation of its edges. Vertices are players and each edge is a game, directed toward the winner. In this talk, we will discuss some recent results on random tournaments. With David Aldous (Berkeley) we construct random tournaments using Strassen’s coupling theorem, yielding a probabilistic proof of Moon’s classical theorem. With Mario Sanchez (Cornell) we study the geometry of random tournaments, with its connections to permutahedra, zonotopes, etc. We show that the recent Coxeter permutahedra are related to tournaments that involve collaboration (and competition, as usual) answering a question of Stanley. Finally, we settle a conjecture of Takács about the asymptotic number of score sequences. The proof involves combinatorics (Erdős–Ginzburg–Ziv numbers), renewal theory and infinitely divisible distributions.