Metal and alloy nanomaterials have intriguing oxidase- and superoxide dismutation-like (SOD-like) activities. However, origins of these activities remain to be studied. Using density functional theory (DFT) calculations, we investigate mechanisms of oxidase- and SOD-like properties for metals Au, Ag, Pd and Pt and alloys Au4-xMx (x = 1, 2, 3; M = Ag, Pd, Pt). We find that the simple reaction-dissociation of O2-supported on metal surfaces can profoundly account for the oxidase-like activities of the metals. The activation (Eact) and reaction energies (Er) calculated by DFT can be used to effectively predict the activity. As verification, the calculated activity orders for series of metal and alloy nanomaterials are in excellent agreement with those obtained by experiments. Briefly, the activity is critically dependent on two factors, metal compositions and exposed facets. On the basis of these results, an energy-based model is proposed to account for the activation of molecular oxygen. As for SOD-like activities, the mechanisms mainly consist of protonation of O2(•-) and adsorption and rearrangement of HO2(•) on metal surfaces. Our results provide atomistic-level insights into the oxidase- and SOD-like activities of metals and pave a way to the rational design of mimetic enzymes based on metal nanomaterials. Especially, the O2 dissociative adsorption mechanism will serve as a general way to the activation of molecular oxygen by nanosurfaces and help understand the catalytic role of nanomaterials as pro-oxidants and antioxidants.