  We present an overview of the microscopic theory of the Dzyaloshinskii-Moriya
(DM) coupling and related exchange-relativistic effects such as exchange
anisotropy, electron-nuclear antisymmetric supertransferred hyperfine
interactions, antisymmetric magnetogyrotropic effects, and antisymmetric
magnetoelectric coupling in strongly correlated 3$d$ compounds focusing on
orthoferrites RFeO$_3$ (R is a rare-earth ion or yttrium Y). % Particular
emphasis will be placed on the role of crystal field effects, the sense and
direction of the Dzyaloshinskii vector, its dependence on the superexchange
bonding geometry, and other subtle features of exchange-relativistic effects.
Most attention in the paper centers around the derivation of the Dzyaloshinskii
vector, its value, orientation, and sense (sign) under different types of the
(super)exchange interaction and crystal field. Microscopically derived
expression for the dependence of the Dzyaloshinskii vector on the superexchange
geometry allows one to find all the overt and hidden canting angles in
orthoferrites RFeO$_3$ as well as corresponding contribution to magnetic
anisotropy. Being based on the theoretical predictions regarding the sign of
the Dzyaloshinskii vector we have predicted and study in detail a novel
magnetic phenomenon, {\it weak ferrimagnetism} in mixed weak ferromagnets with
competing signs of the Dzyaloshinskii vectors. The ligand NMR measurements in
weak ferromagnets are shown to be an effective tool to inspect the effects of
DM coupling in an external magnetic field. Along with orthoferrites RFeO$_3$
and weak ferrimagnets RFe$_{1-x}$Cr$_x$O$_3$, although to a lesser extent, we
address such typical weak ferromagnets as $\alpha$-Fe$_2$O$_3$, FeBO$_3$, and
FeF$_3$.
