Moser's elliptic and parabolic Harnack inequalities are among the most important known results for elliptic operators in divergence form. Roughly speaking, these give pointwise estimates for nonnegative harmonic and parabolic functions. Although these results extend to nice manifolds and nice metric spaces, as soon as one wants to consider domains that have any kind of fractal structure, the proof breaks down. I will talk about joint work with Martin Barlow concerning these Harnack inequalities. For technical convenience we worked with random walks on graphs. We have two main results. The first says that if one has two random walks on a graph whose Dirichlet forms are comparable and one satisfies a parabolic Harnack inequality, then the other will too. Our second result gives necessary and sufficient conditions for the parabolic Harnack inequality to hold for a random walk on a graph.