In proton-exchange-membrane (PEM) fuel cells, electrochemical oxygen reduction reaction (ORR) on the cathode is a critical step at which large energy loss occurs. Theoretical tools at different scales are discussed in this thesis in order to find ORR catalysts with both higher activity and better durability than current Pt and Pt alloys. For catalytic activity, a relatively simple model reaction, ORR by hydrogen molecule under ultra-high-vacuum (UHV) conditions, is studied by first-principles methods on various metallic surfaces, which shows that good catalytic activities of Pt and its alloys originate from moderate adsorption strengths for atoms and molecules involved in the reaction. Then first-principles methods are also applied to study the reaction mechanisms of electrochemical ORR: detailed analyses in the electronic structures of ORR intermediates confirm that all the electron transfers in ORR occur through proton-coupled electron transfer (PCET) mechanism, which is accomplished by proton transfer along hydrogen-bond network from hydrated proton (hydronium) to ORR intermediates on the surface. Furthermore, first-principles methods are also used to search and design new alloy surfaces with optimal activity based on a simple kinetic model. However, the inaccuracy of this simple model makes a comprehensive multiscale ORR model necessary. Thus, a reaction network of ORR elementary steps on limited surface sites is built, and the steady-state solutions provide current density j at given electrode potential U; then a multiscale model of electrode-electrolyte interfacial structure is proposed for function U (σM), where σM is excess surface electron density on metallic electrode; finally, we discuss the principles to achieve a self-consistent multiscale ORR model to output both U and current I to the external circuit. For the stability of Pt as catalyst, we study its surface oxide formation and surface adatom diffusion, which result in the corrosion and coarsening of Pt nanocrystals respectively. It is found that different anti-corrosion stabilities of Pt facets can be explained by their maximum abilities to keep oxygen atoms adsorbed on the top surface layer, and Pt adatom diffusion barriers change with the surface coverage conditions at different U.