The coxeter and weyl Packages. The coxeter package contains 38 Maple procedures for working with root systems and finite Coxeter groups. It provides facilities for manipulating roots, reflections, reduced expressions, for generating permutation representations and irreducible characters of finite Coxeter groups, and for retrieving information such as one finds in the appendices of Bourbaki. The weyl subpackage provides 12 additional procedures for manipulating weights and characters of irreducible representations of semisimple Lie algebras, including functions for computing weight multiplicities, tensor product decompositions, and branching.