Colloidal particles in a liquid crystal (LC) behave very differently from their counterparts in isotropic fluids. Elastic nature of the orientational order and surface anchoring of the director cause long-range anisotropic interactions and lead to the phenomenon of levitation. The LC environment enables new mechanisms of particle transport that are reviewed in this work. Among them the motion of particles caused by gradients of the director, and effects in the electric field: backflow powered by director reorientations, dielectrophoresis in LC with varying dielectric permittivity and LC-enabled nonlinear electrophoresis with velocity that depends on the square of the applied electric field and can be directed differently from the field direction.