Combining first-principles calculations and in situ photoelectron spectroscopy, we show how the composition and structure of the surface of an alloy catalyst is affected by the temperature and pressure of the reagents. The Ag-Cu alloy, recently proposed as an improved catalyst for ethylene epoxidation, forms a thin Cu-O surface oxide, while a Ag-Cu surface alloy is found not to be stable. Several possible surface structures are identified, among which the catalyst surface is likely to dynamically evolve under reaction conditions.