An overview of welding methods and process parameters and its effects on mechanical behaviour and structural integrity of magnesium and its alloys are discussed. These alloys are less dense and beneficial structural alloys for improved energy efficiency, eco-friendliness and driver of circular economic model for sustainable design and innovative ecosystem. While the application of Mg-alloys is projected to increase, understanding the mechanical behaviour and structural integrity of welded joints are critical. Thus, fusion and solid-state welding processes of these alloys are discussed with emphasis on mechanical characterization. Laser welding is the most effective fusion welding technique for most Mg alloys whereas, the predominant solid-state method is friction stir welding. The importance of process variables such as heat inputs, welding velocity (speed) and post weld treatments on the microstructural evolution, on mechanical and physical properties of the distinct zones of the weld joints are described. The weldment is the most susceptible to failure due to phase transformation, defects such as microporosity and relatively coarse grain sizes after solidification. The implication of the design of quality weld joints of Mg alloys are explored with areas for future research directions briefly discussed.