In this talk, we consider the model of random walks in a balanced random environment (RWBRE), a Markov chain that has zero effective speed, and its relation to the stochastic homogenization of non-divergence form difference equations, which concerns the convergence of the random difference equation to a deterministic differential equation. We will survey some recent results on the diffusive behavior of the RWBRE and quantitative homogenization estimates obtained by the speaker and coauthors N. Berger (TU-Munich), M. Cohen (Jerusalem), J-D Deuschel (TU-Berlin), J. Peterson (Purdue), and H.V. Tran (Madison).