We investigate the morphology and kinetics of microphase separation in rod-coil diblock copolymers where each chain consists of a stiff rod block and a flexible coil block. A simplified phenomenological model system is introduced, which is coarse grained in terms of the local concentration difference between the two blocks and the local director field of the rod part. Computer simulations of this set of time-evolution equations in two dimensions reveal that the elastic energy in the rod-block-rich domains affects drastically the domain structures formed in the course of microphase separation. The effects of the external electric field are investigated to examine the domain stability and control the domain morphology. Our recent study of morphological transitions in three dimensions for coil-coil copolymers is also briefly described. © IOP Publishing Ltd.