The PUMAS library is a transport engine for muon and tau leptons in matter. It can operate with a configurable level of details, from a fast deterministic CSDA mode to a detailed Monte Carlo simulation. A peculiarity of PUMAS is that it is revertible, i.e. it can run in forward or in backward mode. Thus, the PUMAS library is particularly well suited for muography applications. In the present document, we provide a detailed description of PUMAS, of its physics and of its implementation. Program Title: The PUMAS library CPC Library link to program files:https://doi.org/10.17632/dnry8jwkd7.1 Developer's repository link:https://niess.github.io/pumas-pages Code Ocean capsule:https://codeocean.com/capsule/4593807 Licensing provisions: LGPL-3.0 Programming language: C99 Nature of problem: Transport of high energy muon or tau leptons in matter. Solution method: Transport engine with a configurable level of details, from a fast deterministic CSDA mode to a detailed Monte Carlo simulation. The transport engine can operate in both forward and backward modes.