Precise structural data are necessary to specify the peculiarities observed in the vicinity of the structural transitions detected in the Bi1-xRExFeO3 compounds. The proposed research will permit to reveal the fundamental factors which stimulate and/or induce the structural transitions, will help to estimate possible presence of intermediate phases. The factors causing low stability of the structural state near the phase boundary will be thoroughly analysed as they are assumed to be the primary reasons of the enhanced physical properties of these compounds. The investigations will enrich materials science with new knowledge about the crystal structure peculiarities and the background of enhanced physical properties of the lead-free multiferroics.