Direct methanol fuel cells are a key enabling technology for clean energy conversion. Using density functional theory calculations, we study the methanol oxidation reaction on model electrodes. We discuss trends in reactivity for a set of monometallic and bimetallic transition metal surfaces, flat and stepped, which includes platinum-based alloys with ruthenium, tin, and copper, as well as nonprecious alloys, overlayer structures, and modified edges. A common lower bound on the overpotential is estimated (ca. 0.3 V). A model for bifunctional alloys is employed to investigate the nature of the active sites on the surface and to screen for novel bimetallic surfaces of enhanced activity. We suggest platinum copper surfaces as promising anode catalysts for direct methanol fuel cells.