  In this report we review the method of explicit calculations of interatomic
exchange interactions of magnetic materials. This involves exchange mechanisms
normally referred to as Heisenberg exchange, Dzyaloshinskii-Moriya interaction
and anisotropic symmetric exchange. The connection between microscopic theories
of the electronic structure, such as density functional theory or dynamical
mean field theory, and interatomic exchange, is given in detail. The different
aspects of extracting information for an effective spin Hamiltonian that
involves thousands of atoms, from electronic structure calculations considering
significantly fewer atoms (1-50) is highlighted. Examples of exchange
interactions of a large group of materials is presented, which involves heavy
elements of the 3d period, alloys between transition metals, Heusler compounds,
multilayer systems as well as overlayers and adatoms on a substrate, transition
metal oxides, 4f elements, magnetic materials in two dimensions and molecular
magnets. Where possible, a comparison to experimental data is made, that
naturally becomes focused on the magnon dispersion. The influence of relativity
is reviewed for a few cases, as is the importance of dynamical correlations.
Development to theories that handle out of equilibrium conditions is also
described here. The review ends with a short description of extensions of the
theories behind explicit calculations of interatomic exchange, to non-magnetic
situations, e.g. that describe chemical (charge) order and superconductivity.
