Density functional theory is used for the evaluation of surface segregation, trends for dissolution of Pt surface atoms in acid medium, and oxygen reduction reaction activity of core-shell materials, containing a monolayer of platinum over a monometallic or bimetallic core. Two groups of cores are investigated: Pt/X with X = Ir, Au; Pd, Rh, Ag; Co, Ni, Cu; and Pt/Pd(3)X, with X = Co, Fe, Cr, V, Ti, Ir, Re. It is found that all the 4d and 5d pure cores may serve as stable cores, and their beneficial effect on the Pt monolayer may be further tuned by alloying the core to another element, here chosen from 3d or 5d groups. The Pd(3)X cores enhance the stability of the surface Pt atoms both in vacuum and under adsorbed oxygen; however the high oxygen philicity of some of the X elements induces their surface segregation that may cause surface poisoning with oxygenated species and their dissolution in acid medium.