Magnesium alloys are widely employed in various applications due to their high strength-to-weight ratio and superior mechanical properties as compared to unalloyed Magnesium. Alloying is considered an important way to enhance the strength of the metal matrix composite but it significantly influences the damping property of pure magnesium, while controlling the rate of corrosion for Mg-based material remains critical in the biological environment. Therefore, it is essential to reinforce the magnesium alloy with a suitable alloying element that improves the mechanical characteristics and resistance to corrosion of Mg-based material. Biocompatibility, biodegradability, lower stress shielding effect, bio-activeness, and non-toxicity are the important parameters for biomedical applications other than mechanical and corrosion properties. The development of various surface modifications is also considered a suitable approach to control the degradation rate of Mg-based materials, making lightweight Mg-based materials highly suitable for biomedical implants. This review article discusses the various binary and ternary Mg alloys, which are mostly composed of Al, Ca, Zn, Mn, and rare earth (RE) elements as well as various non-toxic elements which are Si, Bi, Ag, Ca, Zr, Zn, Mn, Sr, Li, Sn, etc. The effects of these alloying elements on the microstructure, the mechanical characteristics, and the corrosion properties of Mg-based materials were analyzed. The mechanical and corrosion behavior of Mg-based materials depends upon the percentage of elements and the number of alloying elements used in Mg. The outcomes suggested that ZEK100, WE43, and EW62 (Mg-6% Nd-2% Y-0.5% Zr) alloys are effectively used for biomedical applications, having preferable biodegradable, biocompatible, bioactive implant materials with a lower corrosion rate.