Evolutionary game theory introduces time into classical game theory, and can be considered in the biological context of populations playing against each other while changing strategies. The replicator equation proposed by Taylor and Jonker in 1978 is the most commonly used in evolutionary games, although alternatives exist. Such equations are consistent with a fitness function defined as the expected payoff for each strategy, and typically have a nonlinear dependence on the population densities. We consider partial differential equations (PDEs) describing populations playing two-player symmetric games, with both diffusion and fitness gradient migration terms. We find numerically that the fitness gradient flux alters the 1D Hutson-Vickers travelling wave solutions, and leads to spatially structured, stable coexistence states for the prisoner's dilemma. We present numerical simulations and existence results for two-strategy games in the "fitness gradient equation", in which migration is due exclusively to the fitness gradient flux, without replicator dynamic or diffusion.