This paper is a review of the International Reference Ionosphere (IRI) project and model. IRI is recognized as the official standard for the Earth's ionosphere by the International Standardization Organization, the International Union of Radio Science, the Committee on Space Research, and the European Cooperation for Space Standardization. As requested by these organizations, IRI is an empirical (data-based) model representing the primary ionospheric parameters based on the long data record that exists from ground and space observations of the ionosphere. The core model describes monthly averages of the electron density, electron temperature, ion temperature, and ion composition globally in the altitude range from 60 to 2,000 km. Over time additional parameters were added in response to requests from the user community, this includes the equatorial ion drift, the occurrence probability of spread-F and of an F1 layer, auroral boundaries and the electron content from the bottom of the ionosphere to user-specified altitude. IRI has undergone extensive validations and is used for a wide range of applications in science, engineering, and education. This review is the result of many requests we have received for a comprehensive description of the model. It is also meant as a guide for users who are interested in a deeper understanding of the model architecture and its mathematical formalism.