This chapter provides a brief summary of the quantum theory of angular momentum, which is central to the theory of magnetism. It explains that motional angular momentum is insufficient and introduces spin angular momentum. The chapter develops operator techniques to express angular momentum or spin operators in terms of more fundamental fermion or boson operators. Spin-one-half and spin-one are treated separately for use in subsequent chapters on magnetism. In the absence of external forces, the classical angular momentum of a point particle is constant and given by L = r × p. In quantum theory, the analogous quantity, called kinetic angular momentum, is an operator: L = r × (ħ/i)∇. For a spherically symmetric wavefunction, the kinetic angular momentum is zero. However, for functions with nontrivial angular dependence, the kinetic angular momentum is non-zero, requiring the solution of an eigenvalue problem. The components of L do not commute, so they cannot be simultaneously specified. The commutator [Lx, Ly] = iħLz, and similarly for other components, leading to the relation L × L ≡ iħL. This relation defines angular momentum as a vector operator that satisfies this condition. An example shows that the negative of an angular momentum is not an angular momentum, reflecting the pseudovector nature of classical angular momentum. The kinetic angular momentum is expressed in spherical coordinates, with Lz = (ħ/i)∂/∂φ and L+ = ħe^{iφ}(∂/∂θ + i cotθ ∂/∂φ).This chapter provides a brief summary of the quantum theory of angular momentum, which is central to the theory of magnetism. It explains that motional angular momentum is insufficient and introduces spin angular momentum. The chapter develops operator techniques to express angular momentum or spin operators in terms of more fundamental fermion or boson operators. Spin-one-half and spin-one are treated separately for use in subsequent chapters on magnetism. In the absence of external forces, the classical angular momentum of a point particle is constant and given by L = r × p. In quantum theory, the analogous quantity, called kinetic angular momentum, is an operator: L = r × (ħ/i)∇. For a spherically symmetric wavefunction, the kinetic angular momentum is zero. However, for functions with nontrivial angular dependence, the kinetic angular momentum is non-zero, requiring the solution of an eigenvalue problem. The components of L do not commute, so they cannot be simultaneously specified. The commutator [Lx, Ly] = iħLz, and similarly for other components, leading to the relation L × L ≡ iħL. This relation defines angular momentum as a vector operator that satisfies this condition. An example shows that the negative of an angular momentum is not an angular momentum, reflecting the pseudovector nature of classical angular momentum. The kinetic angular momentum is expressed in spherical coordinates, with Lz = (ħ/i)∂/∂φ and L+ = ħe^{iφ}(∂/∂θ + i cotθ ∂/∂φ).