  Dzyaloshinskii-Moriya interaction, DMI in short, represents an antisymmetric
type of magnetic interactions that favour orthogonal orientation of spins and
competes with Heisenberg exchange. Being introduced to explain weak
ferromagnetism in antiferromagnets without an inversion center between magnetic
atoms such an anisotropic interaction can be used to analyze other non-trivial
magnetic structures of technological importance including spin spirals and
skyrmions. Despite the fact that the corresponding DMI contribution to the
magnetic energy of the system has a very compact form of the vector product of
spins, the determination of DMI from first-principles electronic structure is a
very challenging methodological and technical problem whose solution opens a
door into the fascinating microscopic world of complex magnetic materials. In
this paper we review a few such methods developed by us for calculating DMI and
their applications to study the properties of real materials.
