Additive manufacturing technologies, depending on melting of metal powder particles layer upon layer for building 3D functional objects, have widened their application area within biomedical, automotive, aerospace and die/mold industries. Along with the increased application of additive manufacturing, a better understanding of different aspects for the technique has become necessary to fulfill the high demands of quality and productivity. One of the significant process limits of additive manufacturing technologies are residual stresses induced by heating the fine metal powder to the melting point and sudden cooling to the initial temperature. Process simulations performed in a computer environment is of critical importance to predict and to prevent this risk. This paper represents a systematical literature review on the numerical simulations of laser powder bed fusion additive manufacturing. It covers the background and development of analytical models with the contributions of previous ones used for traditional manufacturing techniques such as welding. The details of the numerical modeling studies are provided for thermal and mechanical analysis with the arrangements and assumptions made to reduce the simulation times. Application features are also referred like part material, part size, validation case and the software used for simulations. As a result of these, gap analysis are conducted, needs are established and future research opportunities are expressed.