Inductive inductive definitions is a concept of forming a set inductively while defining a second set depending on it inductively. Variants of it occur frequently in ordinary mathematics. For instance the typical definition of an ordinal notation system defines a set of ordinals inductively while defining a binary relation (<) inductively. Another example are Conway's surreal numbers. A standard version occurs when defining the syntax of type theory inside type theory. We introduce this concept, look at how to formalise it using finitely many rules, and discuss what is known about its proof theoretic strength.