In the process of describing systems and their properties, users of proof assistants introduce not only recursive datatypes, but also sets constructed by a recursive process. These sets are often represented by their characteristic predicates, and we use the terms interchangeably. Such sets or predicates are said to be inductively defined and are found throughout mathematics and computer science. The concept of inductive definition is inherently constructive, and it is equally important to intuitionistic and classical logic. For that reason, inductive predicates can be defined in nearly all proof assistants. Nevertheless, there are some differences in approach. Below we present some representative examples of inductive definitions, outline the theory, discuss some pragmatic issues concerned with implementations and finally look at a few applications.