Hamilton-Jacobi equations form a broad class of first-order partial differential equations. Some such equations model physical phenomena, such as the evolution of a system of particles according to classical mechanics, or the combustion of a flammable gas. On the other hand, any Hamilton-Jacobi equation can be viewed as the evolution of the value function of a two-player differential game. When obtaining sharp quantitative estimates, the viewpoint of differential games (or, in the convex setting, optimal control) is central to our approach. This thesis is primarily concerned with the homogenization, or large-scale behavior, of such equations when the underlying environment exhibits small-scale structure, which we model by either periodicity or randomness. In the periodic setting, we investigate the homogenization rate, which we prove depends on the convexity (or lack thereof) of the Hamiltonian. In the random setting, we focus on the G equation, a convex but noncoercive equation which models combustion. In general, noncoercive equations (and even coercive nonconvex equations in a stationary ergodic enviroment) may not homogenize. However, the G equation is coercive in expectation, which we show is sufficent to analyze the large-scale structure of solutions. In the case of the G equation, our quantitative approach allows us to prove new qualitative results, such as the continuous dependence of the effective Hamiltonian on the law of the environment, and stochastic homogenization when the environment is compressible.