Differential games, as an offspring of game theory and optimal control, provide the modeling and analysis of conflict in the context of a dynamical system. Computing Nash equilibria is one of the core objectives in differential games, with a major bottleneck coming from the notorious intractability of N-player games, also known by the curse of dimensionality. To overcome this difficulty, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms, for which we refer as deep fictitious play. The resulted deep learning algorithm is scalable, parallelizable and model-free. We illustrate the performance of proposed algorithms by comparing them to the closed-form solution of the linear-quadratic game. We also prove the convergence of the fictitious play under appropriate assumptions and verify that the convergent limit forms an open-loop Nash equilibrium. Based on the formulation by backward stochastic differential equations, we extend the strategy of deep fictitious play to compute closed-loop Markovian Nash equilibrium for both homogeneous and heterogeneous large N-player games.