Let \(T\) be a complete first order theory, and \(P\) a distinguished unary predicate in its vocabulary. One can ask: to what extent does the \(P\)-part of a model \(M\) of \(T\) determines \(M\)? If \(P\) always determines \(M\) uniquely, we say that \(T\) is categorical over \(P\). More generally, to what extent is the class of models of \(T\) with a fixed \(P\)-part "classifiable" (that is, can it be classified using a small number of invariants)? Model theory over a predicate is quite unique. On the one hand, it allows to study interesting and natural examples of non-elementary classes using methods and tools from classical model theory. On the other hand, it often exhibits behavior and presents a set of problems more characteristic for non-elementary classes, sometimes requiring a more abstract approach, and some set-theoretic techniques.

In this relatively informal introductory talk, I hope to describe the general framework, introduce some basic concepts (such as stability over a predicate), discuss a few motivating examples, including \(\mathsf{ACFA}_0\) (algebraically closed fields with a generic automorphism) and \(\mathsf{ECF}_0\) (exponentially closed fields), and conclude with some results (old and recent), applications, questions, and conjectures.