After a qualitative discussion of the advantages of the density matrix and of the different ways to introduce it (the statistical, quantum mechanical and operational methods of approach), § 2 deals with the general properties of the density matrix, including a discussion of pure cases and mixtures. A brief discussion is given of Green function techniques and of the relation between Green functions and correlation functions. A discussion of recent developments in the evaluation of partition functions concludes the first part of this article dealing with the theory of density matrix techniques. Sections 5 to 9 discuss applications. The first application is the quantum-chemical one to many-body systems in their ground state, that is, systems at absolute zero, and it is shown how the density matrix fits into the Hartree-Fock and Thomas-Fermi schemes. A brief discussion is given of the theory of diamagnetism. This is followed by a discussion of non-equilibrium processes and of Kubo's approach to transport theory. After that the polarization of beams of electrons or of photons is discussed and it is indicated how density matrix techniques can be used to treat scattering processes. Section 9 concludes this part of the paper by a brief account of density matrix theory applications to resonance and relaxation phenomena. Finally, the theory of measurement in quantum mechanics is considered.