Metal additive manufacturing (MAM) has experienced huge growth thanks to the increase in production performance as well as our fundamental knowledge of the process. Production defects closely linked to the multiphysics taking place, however, slow down the implementation of components produced with this technology. Hence, this talk will present an overview of the most well-known numerical procedures for multi-physics modelling of metal additive manufacturing processes with special focus on the Laser Powder Bed Fusion (L-PBF) as well as Directed Energy Deposition (DED) processes. While doing so, they will be classified not only based on their length-scale as often seen, but also based on the involved physics, as well as the coupling strategies at both the deposition-scale and part-scale. Specific phenomena such as marangoni convection, capillary forces, solidification, laser and material interaction as well as evaporation are considered. Selected examples of varying complexity from simpler models to full-blown multi-physics models will be presented. These include simulation of deposition scale phenomena such as lack of fusion voids, key-hole induced porosities, spatter and denudation as well as part scale residual stresses and deformations.