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 3d compounds focusing on orthoferrites RFeO3 (R is a rare-earth ion or yttrium Y). 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 RFeO3 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, 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 RFeO3 and weak ferrimagnets RFe1 -xCrxO3, although to a lesser extent, we address such typical weak ferromagnets as α-Fe2O3, FeBO3, and FeF3.