Induction-recursion

In intuitionistic type theory (ITT), a discipline within mathematical logic, induction-recursion is a feature for simultaneously declaring a type and function on that type. It allows the creation of larger types, such as universes, than inductive types. The types created still remain predicative inside ITT.
An inductive definition is given by rules for generating elements of a type. One can then define functions from that type by induction on the way the elements of the type are generated. Induction-recursion generalizes this situation since one can "simultaneously" define the type and the function, because the rules for generating elements of the type are allowed to refer to the function.
Induction-recursion can be used to define large types including various universe constructions. It increases the proof-theoretic strength of type theory substantially. Nevertheless, inductive-recursive recursive definitions are still considered predicative.
Background.
Induction-Recursion came out of investigations to the rules of Martin-Löf's intuitionistic type theory. The type theory has a number of "type formers" and four kinds of rules for each one. Martin-Löf had hinted that the rules for each type former followed a pattern, which preserved the properties of the type theory (e.g., strong normalization, predicativity). Researchers started looking for the most general description of the pattern, since that would tell what kinds of type formers could be added (or not added!) to extend the type theory.
The "universe" type former was the most interesting, because when the rules were written "à la Tarski", they simultaneously defined the "universe type" "and "a function that operated on it. This eventually lead Dybjer to Induction-Recursion.
Dybjer's initial papers called Induction-Recursion a "schema" for rules. It stated what type formers could be added to the type theory. Later, he and Setzer would write a new type former with rules that allowed new Induction-Recursion definitions to be made inside the type theory. This was added to the Half proof assistant (a variant of Alf).
The idea.
Before covering Inductive-Recursive types, the simpler case is Inductive Types. Constructors for Inductive types can be self-referential, but in a limited way. The constructor's parameters must be "positive": 
With Inductive types, a parameter's type can depend on earlier parameters, but they cannot refer to ones of the type being defined. Inductive-Recursive types go further and parameter's types "can" refer to earlier parameters that use the type being defined. These must be "half-positive":
So, if formula_1 is the type being defined and formula_2 is the function being (simultaneously) defined, these parameter declarations are positive:
This is half-positive:
These are not positive nor half-positive:
Universe example.
A simple common example is the Universe à la Tarski type former. It creates a type formula_23 and a function formula_24. There is an element of formula_23 for every type in the type theory (except formula_23 itself!). The function formula_24 maps the elements of formula_23 to the associated type.
The type formula_23 has a constructor (or introduction rule) for each type former in the type theory. The one for dependent functions would be:
formula_30
That is, it takes an element formula_31 of type formula_23 that will map to the type of the parameter, and a function formula_33 such that for all values formula_34, formula_35 maps to the return type of the function (which is dependent on the value of the parameter, formula_36). (The final formula_37 says that the result of the constructor is an element of type formula_23.)
The reduction (or computation rule) says that
formula_39 becomes formula_40.
After reduction, the function formula_24 is operating on a smaller part of the input. If that holds when formula_24 is applied to any constructor, then formula_24 will always terminate. Without going into the details, Induction-Recursion states what kinds of definitions (or rules) can be added to the theory such that the function calls will always terminate.
Usage.
Induction-Recursion is implemented in Agda and Idris.