The thrust of Professor Schwarz's work in mathematical physics has been to apply topology to physics. In this aim he anticipated some of the work of the physicist Edward Witten. His most important result, joint with Belavin, Polyakov, and Tyupkin, was the discovery of instantons, which are solutions to classical gauge field equations that are localized in four Euclidean dimensions. Although they are Euclidean objects, instantons manifest themselves in 4-dimensional Minkowskian spacetime as avenues for quantum tunneling. In this guise, they were used by t'Hooft to arrive at one of the notorious predictions of a Grand Unified Theory of physics (excluding only gravity): that the proton must be unstable. Finally, instantons reappeared in pure mathematics in Donaldson theory, a body of ideas which applies gauge field theory techniques to demonstrate many surprising non-trivialities about smooth 4-manifolds.