PRECESSION is a comprehensive toolbox for exploring the dynamics of precessing black-hole binaries in the post-Newtonian regime. It allows study of the evolution of the black-hole spins along their precession cycles, performs gravitational-wave-driven binary inspirals using both orbit-averaged and precession-averaged integrations, and predicts the properties of the merger remnant through fitting formulas obtained from numerical-relativity simulations. PRECESSION can add the black-hole spin dynamics to larger-scale numerical studies such as gravitational-wave parameter estimation codes, population synthesis models to predict gravitational-wave event rates, galaxy merger trees and cosmological simulations of structure formation, and provides fast and reliable integration methods to propagate statistical samples of black-hole binaries from/to large separations where they form to/from small separations where they become detectable, thus linking gravitational-wave observations of spinning black-hole binaries to their astrophysical formation history. The code is also useful for computing initial parameters for numerical-relativity simulations targeting specific precessing systems.