Giant radio sources (GRSs) are extremely valuable for better understanding the last stages of evolution of radio sources and their host galaxies, constraining the analytical models of their time evolution, verifying orientation-dependent unified schemes, etc. On the basis of the population statistics, derived from the existing evolutionary models, we have realized a deep deficit of low-luminosity GRSs observed at redshifts < 0.3–0–4 (Machalski and Jamrozy, 2000). In order to determine the cause of this deficit and find the missing giants, we have selected a sample of faint FRII-type GRS candidates, using the VLA 1.4 GHz sky surveys: NVSS (Condon et al., 1998) and FIRST (Becker et al., 1995). The first results of a search for the radio cores of our candidates with deep VLA observations at 4.9 GHz and 8.4 GHz, and the follow-up spectroscopic observations of identified optical counterparts is presented.