Liquid phase electroepitaxy (LPEE) is a low temperature, solution growth technique which has been proven to be successful in growing high quality, thick compound and alloy semiconducting single crystal layers. The availability of such high quality alloy layers would open new horizons in the production of optoelectronic and high-speed devices. Due to the technological importance of this crystal growth technique, some theoretical models for its growth process have been developed recently to provide needed information to experimentalists to develop growth configurations for reproducible desired single crystals. These models have shed light on various aspects of the LPEE growth process. This review article, with 71 references, provides the reader with an overview of recently developed macroscopic continuum models for the LPEE growth process of binary and ternary semiconductors. Fundamental equations of these models are obtained from the basic principles of electrodynamics and thermomechanics of the continuum. The models include various thermoelectric effects observed in LPEE and also incorporates microscopic surface phenomena such as surface kinetics. Results of numerical simulations are presented, and compared with available experimental data. The significance of research results are discussed.