On Provability Logic

This is an introductory paper about provability logic, a modal propositional logic in which necessity is interpreted as formal provability. We discuss the ideas that led to establishing this logic, we survey its history and the most important results, and we emphasize its applications in metamathematics. Stress is put on the use of Gentzen calculus for provability logic. We sketch our version of a decision procedure for provability logic and mention some connections to computational complexity.