The essential role played by differentiable structures in physics is reviewed in light of recent mathematical discoveries that topologically trivial space?time models, especially the simplest one, R4, possess a rich multiplicity of such structures, no two of which are diffeomorphic to each other and thus to the standard one. This means that physics has available to it a new panoply of structures for space?time models. These can be thought of as a source of new global, but not properly topological, features.