In this paper, we study mixed continuous-discrete minimax problems where the minimization is over a continuous variable belonging to Euclidean space and the maximization is over subsets of a given ground set. We introduce the class of convex-submodular minimax problems, where the objective is convex with respect to the continuous variable and submodular with respect to the discrete variable. Even though such problems appear frequently in machine learning applications, little is known about how to address them from algorithmic and theoretical perspectives. For such problems, we first show that obtaining saddle points are hard up to any approximation, and thus introduce new notions of (near-) optimality. We then provide several algorithmic procedures for solving convex and monotone-submodular minimax problems and characterize their convergence rates, computational complexity, and quality of the final solution. Finally, we provide experiments to showcase the effectiveness of our purposed methods on training robust recommender systems in the presence of poisoning attacks.