The arc discharge plasma (ADP) technology has been widely developed in the fields of cutting, welding, spraying and nanomaterials synthesis over the past 20 years. However, during the process of ADP, it is difficult to explain the generation and evolution of arc column, the interaction between arc column and electrodes, as well as the effect of plasma generator structure on the physical characteristics of ADP by experimental means. Therefore, numerical simulation has become an effective mean to explore the physical characteristics of ADP, but also faces severe challenges because it involves multiple physical field coupling, resolution of multiscale features as well as robustness in the presence of large gradients. From the point of view of the construction of ADP mathematical physical models and combined with the practical application of ADP, this paper systematically reviews the researches on physical properties of arc column, near-cathode region, near-anode region as well as the today’s state of the numerical simulation of plasma generators. It provides a good reference for further mastering the physical characteristics of plasma, guiding the industrial application of plasma and optimizing the design of plasma generators. Meanwhile, the relevant computational aspects are discussed and the challenges of plasma numerical simulation in the future are summarized.