The up-to-date theoretical and experimental data on the fairly-high-velocity motion of a solitary domain wall in a weak ferromagnet are reviewed. The experimental data pertain to orthorhombic and uniaxial crystals of the orthoferrite and iron-borate types. The theoretical analysis demonstrates the great generality of the nonlinear dynamics of walls in weak ferromagnets of different symmetries. The necessary data on the magnetic structure and the magneto-optical properties of weak ferromagnets are presented. The experimental techniques of investigating domain-wall motion, including non-steady-state motion, are described. Mobility and limiting-wall-velocity data are given, and a quantitative theoretical description of these experiments is presented. The magnetoelastic anomalies observed experimentally in the case when the wall velocity is close to the velocity of sound are described. A theory of this phenomenon is given which is based on the notion of Cherenkov emission of sound by a moving wall. The results of the investigation of the non-steady-state and nonlinear domain-wall motion that arises due to an instability of the wall front are presented, and theoretical models are proposed for such motion.