Lithium niobate (LiNbO3) and lithium tantalate (LiTaO3) are among the most important and most widely used materials of coherent and nonlinear optics, as well as acoustics. High degree of uniformity and reproducibility has become the foundation of technology for manufacturing high-quality crystals, absorbed by many suppliers around the world. However, the above areas do not limit the use of LiNbO3 and LiTaO3 due to their unique piezoelectric and ferroelectric properties. One promising application of crystals is the design of electromechanical transducers for precision sensors and actuators. In this respect, the high thermal stability of the piezoelectric and mechanical properties, the lack of hysteresis and creep make it possible to create electromechanical converters with wide operating temperature range, that is beyond the capability of commonly used ferroelectric ceramics. The main advantage of LiNbO3 and LiTaO3 over other single-crystal piezoelectrics is ferroelectric domain structure regulation toward targeted impact on the device characteristics. One of the most striking examples of electromechanical transducer design through domain engineering is the formation of a so-called bidomain ferroelectric structure in crystal. It represents a single-crystalline plate with two macrodomains with opposite directions of spontaneous polarization vectors separated by a charged domain wall. High switching fields make inversion domains stable at temperatures up to 1000 °C. This review summarizes the main achievements in the formation of bidomain structure and near surface inversion domains in LiNbO3 and LiTaO3 crystals. We present the domain structure virtualization methods in crystals and non-destructive methods for controlling the domain boundary position. The report contains a comparative analysis of the methods for forming inversion domains in crystals, and the patterns and technological control methods of the domain structure are discussed. The basic physical models have been proposed in the literature to explain the effect of the inversion domains formation. In the present paper we outline what one sees as strengths and weaknesses of these models. The strategies of crystallographic cut selection to create devices based on bidomain crystals are briefly discussed. We provide examples of the implementation of devices based on bidomain crystals such as actuators, sensors, acoustic transducers, and waste energy collection systems.