<1>¿¿Radiation of an oscillating electric dipole which is travelling with a constant velocity in homogeneous isotropic medium with given dispersion characteristic is studied. The cases of cold plasma and the medium with the dispersion of a resonant type are considered. It is shown that in the case of the resonantly dispersive medium, the radiation spectrum consists (depending on the problem parameters) either of two separated frequency ranges or of one frequency range. Regularities characterizing the dependence of the radiated power on the source motion velocity at various values of the resonant and plasma frequencies are derived. The case when the dipole travels in a moving medium the velocity of which is parallel or antiparallel to the source motion velocity is also considered. It is noted in particular that at some parameters of the problem, the energy loss of the source are negative.