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The Shimura correspondence, going back to the early 1970s, relates two different kinds of objects that from a distance appear almost the same: integral and half integral weight modular forms.  In this talk I will explain what these objects are, why they are fundamentally different, and why a connection between them is of consequence.  Then I will discuss a half-century of generalizations, including on-going joint work with Omer Offen about a new Shimura correspondence and a new relative trace formula.  This work uses in a critical way an identity due to Duke and Iwaniec.