Except for special classes of games, there is no systematic framework for analyzing the dynamical properties of multi-agent strategic interactions. Potential games are one such special but restrictive class of games that allow for tractable dynamic analysis. Intuitively, games that are “close” to a potential game should share similar properties. In this paper, we formalize and develop this idea by quantifying to what extent the dynamic features of potential games extend to “near-potential” games. We first show that in an arbitrary finite game, the limiting behavior of better-response and best-response dynamics can be characterized by the approximate equilibrium set of a close potential game. Moreover, the size of this set is proportional to a closeness measure between the original game and the potential game. We then focus on logit response dynamics, which induce a Markov process on the set of strategy profiles of the game, and show that the stationary distribution of logit response dynamics can be approximated using the potential function of a close potential game, and its stochastically stable strategy profiles can be identified as the approximate maximizers of this function. Our approach presents a systematic framework for studying convergence behavior of adaptive learning dynamics in finite strategic form games.