Additive Manufacturing (AM) is a process that fabricates products by manufacturing materials according to a three-dimensional model. It has recently gained attention due to its environmental advantages, including reduced energy consumption and high material utilization rates. However, controlling defects such as melting issues and residual stress, which can occur during metal additive manufacturing, poses a challenge. The trial-and-error verification of these defects is both time-consuming and costly. Consequently, efforts have been made to develop phenomenological models that understand the influence of process variables on defects, and mechanical/ electrical/thermal properties of geometrically complex products. This paper introduces modeling techniques that can simulate the powder additive manufacturing process. The focus is on representative metal additive manufacturing processes such as Powder Bed Fusion (PBF), Direct Energy Deposition (DED), and Binder Jetting (BJ) method. To calculate thermal-stress history and the resulting deformations, modeling techniques based on Finite Element Method (FEM) are generally utilized. For simulating the movements and packing behavior of powders during powder classification, modeling techniques based on Discrete Element Method (DEM) are employed. Additionally, to simulate sintering and microstructural changes, techniques such as Monte Carlo (MC), Molecular Dynamics (MD), and Phase Field Modeling (PFM) are predominantly used.