We study mean field games with scalar Itô-type dynamics and costs that are

submodular with respect to a suitable order relation on the state and measure space. The

submodularity assumption has a number of interesting consequences. Firstly, it allows us to

prove existence of solutions via an application of Tarski's fixed point theorem, covering cases

with discontinuous dependence on the measure variable. Secondly, it ensures that the set

of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal

solution. Thirdly, it guarantees that those two solutions can be obtained through a simple

learning procedure based on the iterations of the best-response-map. The mean field game

is first defined over ordinary stochastic controls, then extended to relaxed controls. Our

approach allows also to treat a class of submodular mean field games with common noise in

which the representative player at equilibrium interacts with the (conditional) mean of its

state's distribution.