Magnesium (Mg) alloys have received increasing interest in the past two decades as biomaterials due to their excellent biological compatibility. However, the corrosion resistance of Mg alloys is relativity low which limits their usage in degradable implant applications, and controlling the corrosion resistance is the key to solving this problem. This review discusses the relative corrosion mechanisms, including pitting, filiform, high temperature, stress corrosion, etc., of Mg alloys. Various approaches like purification (Fe, Ni, Cu, etc.), micro-alloying (adding Zn, Mn, Ca, RE elements, and so on), grain refinement (severe plastic deformation, SPD, etc.), and surface modifications (various coating methods) to control corrosion and biological performance are summarized. Moreover, the in vivo implantations of Mg alloy vascular stents and the issues that have emerged based on the reports in recent years are introduced. It is recommended that corrosion mechanisms should be further investigated as there is no method that can remove all the impurities and a new purification approach needs to be developed. The concentration of micro-alloy elements should be carefully controlled to avoid superfluous compounds. Developing new continuous SPD methods to achieve fine-grained Mg alloys with a large size scale is necessary. The development of a multifunctional coating could also be considered in controlling the Mg degradation rate. Moreover, the research trends and challenges in the future of Mg biomaterials are proposed.