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Models that have some but not all features of standard quantum theory can be valuable in several ways, as Bell, Ghirardi-Rimini-Weber-Pearle, Hardy, Spekkens and many others have shown. One is to illuminate quantum theory and shed light on possible reaxiomatisations or reformulations. Another is to suggest experiments that might confirm some untested aspect of quantum theory or point the way to a new theory. I discuss here some models that combine quantum theory and gravity and experimental tests.