Magnesium can be reversibly deposited from ethereal solutions of Grignard salts of the RMgX type (R = alkyl, aryl groups, and X = halides: Cl, Br), and complexes of the Mg(AX4-nRn′R′n″)2 type (A = Al, B; X = Cl, Br; R, R′ = alkyl or aryl groups, and n′ + n″ = n). These complexes can be considered as interaction products between R2Mg bases and AX3-nRn Lewis acids. The use of such complexes in ether solvents enables us to obtain solutions of reasonable ionic conductivity and high anodic stability, which can be suitable for rechargeable Mg battery systems. In this paper we report on the study of variety of Mg(AX4-nRn)2 complexes, where A = Al, B, Sb, P, As, Fe, and Ta; X = Cl, Br, and F; and R = butyl, ethyl, phenyl, and benzyl (Bu, Et, Ph, and Bz, respectively) in several solvents, including tetrahydrofuran (THF), 2Me-THF, 1-3 dioxolane, diethyl ether, and polyethers from the "glyme" family, including dimethoxyethane (glyme), (CH3OCH2CH2)2O(diglyme), and CH3(OCH2CH2)4OCH3 (tetraglyme), as electrolyte solutions for rechargeable magnesium batteries. It was found that Mg(AlCl4-nRn′R′n″)2 complexes (R, R′ = Et, Bu and n′ + n″ = n) in THF or glymes constitute the best results in terms of the width of the electrochemical window (>2 V), from which magnesium can be deposited reversibly. These solutions were found to be suitable for use in rechargeable magnesium batteries. A variety of electrochemical and spectroscopic studies showed that these solutions have a complicated structure, which is discussed in this paper. It is also clear from this work that Mg deposition-dissolution processes in these solutions are far from being simple reactions of Mg/Mg+2 redox couple. The conditions for optimized Mg deposition-dissolution processes are discussed herein.