We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L∞⌒BVloc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructingthe entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large L1⌒L∞⌒BVloc perturbation of the Riemann initial data, as longas the corresponding solutions are in LN and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containingvacuum in the class of entropy solutions in LN with arbitrarily large oscillation.