Type theory is a regime for classifying objects (including events) into categories called types. It was originally designed in order to overcome problems relating to the foundations of mathematics relating to Russell’s paradox. It has made an immense contribution to the study of logic and computer science and has also played a central role in formal semantics for natural languages since the initial work of Richard Montague building on the typed λ-calculus. More recently, type theories following in the tradition created by Per Martin-Löf have presented an important alternative to Montague’s type theory for semantic analysis. These more modern type theories yield a rich collection of types which take on a role of representing semantic content rather than simply structuring the universe in order to avoid paradoxes.