This monograph introduces discrete-time mean field games with finite state and action spaces, emphasizing optimization formulations and learning algorithms. Starting from Markov decision processes and finite-player stochastic games, it develops the notion of mean field Nash equilibrium, establishes existence and uniqueness results under suitable assumptions, and explains how mean field equilibria yield approximate equilibria in large finite-player games. Building on the linear programming formulation of Markov decision processes, occupation measures and primal-dual formulations connect equilibrium computation with mathematical optimization. The monograph presents fixed-point iterations, fictitious play, online mirror descent, and optimization-based methods, discussing their convergence properties and limitations. Numerical examples illustrate crowd motion, traffic routing, epidemic management, and the computation of multiple equilibria. It combines examples, theoretical results, and implementable algorithms, with bibliographic notes providing connections to the broader literature.

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