As the lightest structural alloys, magnesium alloys offer significant potential for improving the energy efficiency of various transportation systems. The lack of sufficient weldability of Mg alloys is a crucial barrier to their potential use in safety-critical applications. Porosity formation, grain structure engineering, solidification cracking, liquation, and liquation cracking are the key metallurgical challenges to obtain reliable and robust fusion welds in Mg alloys. This critical review highlights the current understating regarding controlling the metallurgical phenomena during fusion welding of Mg alloys and discusses the unresolved metallurgical challenges to shed light on the path forward to enhance fusion weldability of Mg alloys.