Large differences between the characteristic times of various electronic relaxation processes (momentum relaxation, energy relaxation, intervalley relaxation, electron-hole recombination) make it possible to divide carriers into groups between which relaxation is relatively slow. Each of the ``long'' relaxation times can be matched by a characteristic diffusion length which is much greater than the usual mean free path. Transport coefficients of such groups are generally anisotropic even in cubic crystals and the anisotropy varies from group to group (this anisotropy may be natural or it may be induced by pressure, magnetic field, etc.). Therefore, the passage of a current produces nonequilibrium carrier densities in such groups. The density gradients are oriented at right-angles to the current and they decay over distances of the order of the diffusion length. The effects associated with the formation of nonequilibrium carriers and the influence of their diffusion on the transport coefficients are referred to in the paper as the anisotropic size effects. The paper reviews experimental and theoretical investigations of various manifestations of such effects. An analysis is made of the size dependences of the electrical conductivity and magnetoresistance manifested in ``thick'' samples (thickness of the order of the diffusion length). Other topics considered include nonlinearity of the electrical conductivity in relatively weak fields, redistribution of carriers in ``strong'' fields (accompanied by giant changes in the total number of carriers and by formation of domains, depletion layers, and accumulation layers), influence of the anisotropic size effects on the skin effect (which changes the surface impedance of semimetals by an order of magnitude), and electromagnetic excitation of sound in semimetals.