In this talk, we first propose a new class of metrics and show that under such metrics, the convergence of empirical measures in high dimensions is free of the curse of dimensionality, in contrast to Wasserstein distance. Proposed metrics are the integral probability metrics, where we propose criteria for test function spaces. Examples include RKHS, Barron space, and flow-induced function spaces. One application studies the construction of Nash equilibrium for the homogeneous n-player game by its mean-field limit (mean-field game). Another application is to show the ability to overcome curves of dimensionality of deep learning algorithms, for example, in solving Mckean-Vlasov forward-backward stochastic differential equations with general distribution dependence. The is joint work with Jiequn Han and Jihao Long.