Solving continuous minimax optimization is of extensive practical interest, yet notoriously unstable and difficult. This paper introduces the learning to optimize(L2O) methodology to the minimax problems for the first time and addresses its accompanying unique challenges. We first present Twin-L2O, the first dedicated minimax L2O method consisting of two LSTMs for updating min and max variables separately. The decoupled design is found to facilitate learning, particularly when the min and max variables are highly asymmetric. Empirical experiments on a variety of minimax problems corroborate the effectiveness of Twin-L2O. We then discuss a crucial concern of Twin-L2O, i.e., its inevitably limited generalizability to unseen optimizees. To address this issue, we present two complementary strategies. Our first solution, Enhanced Twin-L2O, is empirically applicable for general minimax problems, by improving L2O training via leveraging curriculum learning. Our second alternative, called Safeguarded Twin-L2O, is a preliminary theoretical exploration stating that under some strong assumptions, it is possible to theoretically establish the convergence of Twin-L2O. We benchmark our algorithms on several testbed problems and compare against state-of-the-art minimax solvers. The code is available at: https://github.com/VITA-Group/L2O-Minimax.