Despite the rich existing literature about minimax optimization in continuous settings, only very partial results of this kind have been obtained for combinatorial settings.In this paper, we fill this gap by providing a characterization of submodular minimax optimization, the problem of finding a set (for either the min or the max player) that is effective against every possible response.We show when and under what conditions we can find such sets.We also demonstrate how minimax submodular optimization provides robust solutions for downstream machine learning applications such as (i) prompt engineering in large language models, (ii) identifying robust waiting locations for ride-sharing, (iii) kernelization of the difficulty of instances of the last setting, and (iv) finding adversarial images. Our experiments show that our proposed algorithms consistently outperform other baselines.