  In this paper, we study harmonic functions on metric measure spaces with
Riemannian Ricci curvature bounded from below, which were introduced by
Ambrosio-Gigli-Savar\'e. We prove a Cheng-Yau type local gradient estimate for
harmonic functions on these spaces. Furthermore, we derive various optimal
dimension estimates for spaces of polynomial growth harmonic functions on
metric measure spaces with nonnegative Riemannian Ricci curvature.
