We provide an abstract framework for submodular mean field games and identify verifiable sufficient conditions that allow to prove existence and approximation of strong mean field equilibria in models, where data may not be continuous with respect to the measure parameter and common noise is allowed. Mean field games (MFGs for short), as introduced by Lasry and Lions and, independently, by Huang, Malhamé and Caines, are limit models for non-cooperative symmetric N-player games with mean field interaction as the number of players N tends to infinity. In this talk, we focus on a lattice-theoretic approach, which is based on first order stochastic dominance, and prove the existence of a fixed point using Tarski's fixed point theorem. The setting is general enough to encompass qualitatively different problems, such as mean field games for discrete time finite space Markov chains, singularly controlled and reflected diffusions, and mean field games of optimal timing. The talk is based on joint work with Jodi Dianetti, Giorgio Ferrari, and Markus Fischer.