We study the best-response dynamics in games with random payoffs. We consider two-player bimatrix games, with $n$ strategies each. We assume that payoffs come from a continuous distribution, and study the best-response dynamics of these games, after imposing some structure on their payoff matrices. In particular, we focus on symmetric games, antisymmetric games, common-interest games, zero-sum games, and games with independent payoffs. When two players take turns in playing the best response to the opponent's action, they eventually end up cycling through the same sequence of actions. The main focus of the paper is the cycle structure of large random games. In particular, we study the number of cycles, the number of cycles of a certain length, the length of the longest cycle, and related quantities. We show that the structure of the game has an important impact on the typical cycle structure. Our arguments are based on a representation of the best-response dynamics of these games as directed bipartite graphs, and on the cycle structure that arises in them.

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