In this book the concept of indistinguishability is defined for identical particles by the symmetry of the state rather than by the symmetry of observables. It applies, therefore, to both the classical and the quantum framework. In this setting the particles of classical Maxwell-Boltzmann statistics are indistinguishable and independent. The author describes symmetric statistical operators and classifies these by means of extreme points and by means of extendibility properties. The three classical statistics are derived in abelian subalgebras. The classical theory of indistinguishability is based on the concept of interchangeable random variables which are classified by their extendibility properties. For the description of infinitely extendible interchangeable random variables de Finetti's theorem is derived and generalizations covering the Poisson limit and the central limit are presented. A characterization and interpretation of the integral representations of classical photon states in quantum optics is derived in abelian subalgebras. Unextendible indistinguishable particles are analyzed in the context of nonclassical photon states. The book addresses mathematical physicists and philosophers of science.