Mean-field game (MFG) is a framework for modeling large systems of interacting agents that play non-cooperative differential games. In a PDE form, an MFG system comprises a Hamilton-Jacobi PDE coupled with a Kolmogorov-Fokker-Planck or continuity equation. Such systems are challenging to analyze and solve due to their non-linear forward-backward structure. I will discuss computational methods and related theoretical aspects of these PDEs. More specifically, I will present a new framework for solving systems with nonlocal interactions and new insights into the variational structure of MFG systems. One of the most efficient techniques for the analysis and numerical solution of MFG systems is the extension of the celebrated Benamou-Brenier formulation of the optimal transportation problem to MFG systems. More specifically, under suitable structural assumptions, the MFG system can be regarded as a system of first-order optimality conditions for convex energy. Such systems are called potential, and extensions of the Benamou-Brenier technique beyond these systems were unknown. I will show that non-potential systems that satisfy the so-called Lasry-Lions monotonicity condition admit a monotone-inclusion version of the Benamou-Brenier formulation. I will present new numerical methods based on this formulation and discuss its possible applications in the analysis of non-potential MFG systems. Finally, I will conclude with a few remarks on the applications of MFG to problems in swarm control and economics.