We develop many-valued logic, including a generic abstract model theory, over a fully abstract syntax. We show that important many-valued logic model theories, such as traditional first-order many-valued logic and fuzzy multi-algebras, may be conservatively embedded into our abstract framework. Our development is technically based upon the so-called theory of institutions of Goguen and Burstall and may serve as a template for defining at hand many-valued logic model theories over various concrete syntaxes or, from another perspective, to combine many-valued logic with other logical systems. We also show that our generic many-valued logic abstract model theory enjoys a couple of important institutional model theory properties that support the development of deep model theory methods.