Theoretical models of misfit stress relaxation in heterogeneous crystalline nanostructures are reviewed in brief. It is shown that the main channel of relaxation is the formation of misfit dislocations. Some mathematical tools for continuum modeling of misfit stress relaxation through generation of discrete dislocations in spherical and cylindrical nanostructures are considered with special attention to the strain energies of the dislocations and the energies of elastic interaction between them. The critical conditions and energy barriers for the formation of prismatic dislocation loops and straight edge misfit dislocations in core-shell nanoparticles and nanowires with various types of cores, in Janus nanoparticles and nanowires, in axially inhomogeneous nanowires with transverse interfaces, and in free-standing composite nanolayers are discussed.