This monograph integrates unitary symmetry and combinatorics, showing in great detail how the coupling of angular momenta in quantum mechanics is related to binary trees, trivalent trees, cubic graphs, MacMahon's master theorem, and other basic combinatorial concepts. A comprehensive theory of recoupling matrices for quantum angular momentum is developed. For the general unitary group, polynomial forms in many variables called matrix Schur functions have the remarkable property of giving all irreducible representations of the general unitary group and are the basic objects of study. The structure of these irreducible polynomials and the reduction of their Kronecker product is developed in detail, as is the theory of tensor operators.Contents:Quantum Angular MomentumComposite SystemsGraphs and Adjacency DiagramsGenerating FunctionsThe Dλ-Polynomials: FormOperator Actions in Hilbert SpaceThe Dλ-Polynomials: StructureThe General Linear and Unitary GroupsTensor Operator TheoryCompendium A: Basic Algebraic ObjectsCompendium B: Combinatorial ObjectsReadership: Graduate students and researchers in physics and mathematics who wish to learn about the relationships between symmetry and combinatorics.