The present status of theoretical studies of radiative electron capture (REC) is discussed. Difficulty in the formulation comes from the fact that we do not have enough knowledge concerning the exact wave functions of three-body scattering states. The validity of the inverse-photoionization concept for REC is examined. A problem occurring in the calculations in the strong potential Born approximation is also discussed. We have made calculations of the total REC cross section in the relativistic impulse approximation including the intermediate process of positron-electron pair creation. Various characteristic features of REC X-rays are discussed in the relativistic impulse approximation.