Abstract: The low-pressure hollow cathode discharge made of a hollow circular tube and an anode is a type of simple structure discharge system. In particular, under the arc discharge mode, hollow cathodes have high plasma density and energy density with a wide range of adaptability of pressure and current. Low-pressure hollow cathode arc (HCA) discharges have been widely used as plasma sources in various fields such as manufacturing, vacuum welding, and aerospace since the 1960s. Despite the early experimental and applied researches on low-pressure HCA discharges, the basic theoretical study was relatively lagged much behind, resulting in many unanswered questions, such as the optimal discharge operating parameters, the power deposition inside the cathode, the causes of plasma instability, and how to effectively reduce cathode erosion and so on. Due to the special discharge structure of the hollow cathode, it is difficult to make an accurate experimental diagnosis, so a reasonable numerical simulation is an effective study method. However, up to now, there is still a lack of complete and effective numerical models which can evaluate various physical fields in the low-pressure hollow cathode discharges. To address the above problems and difficulties, a comprehensive and self-consistent 2D multi-physical coupling numerical model based on a commercial program of finite element method, the COMSOL Multiphysics, was provided in this paper. The model involves plasma transport, arc flow and heat transfer, and cathode thermal equilibrium, and can consider the effect of an applied magnetic field. The processes of secondary electron emission, thermal-field electron emission, ions and backflow high-energy electrons bombardment, and thermal radiation from the cathode surface are considered in the cathode thermal equilibrium process. Based on the above background, this paper works from the following aspects： In Chapter 1, the basic concepts of low-pressure HCA discharge including the hollow cathode effect, the basic characteristics, and operation modes were introduced firstly; Secondly, the application fields, development history, and overseas and domestic research status of hollow cathode discharge were reviewed; finally, the problems were presented and the research background was explained, and the research purpose of this paper was clarified. In Chapter 2, a complete and self-consistent numerical model of low-pressure hollow cathode discharge was proposed based on the fundamental theory and assumptions, and the set of control equations and boundary conditions in the model were elaborated. In addition, the electron energy distribution function, the collision processes, the solving tools of this model, and calculation schemes were introduced in detail. Finally, a validation example was given to test the rationality and applicability of the numerical model. In Chapter 3, the fundamental plasma properties of low-pressure hollow cathode arcs were investigated. Firstly, the ion Joule heating effect was studied. The results showed that the temperature distributions of the arc and cathode are only able to approach the experimental measurements after considering the ion Joule heating, which shows that the Joule heating of ions is crucial for the heating of the arc plasma. Secondly, by comparing the radial distribution of electron and ion density inside the cathode, the structure of the cathode sheath could be simulated well using this model. Finally, it was shown that the thermal radiation from the cathode surface is an important cooling mechanism of the cathode and only under higher surface emissivity can balance the larger heat flow given by the plasma to the cathode, and the temperature distribution of the cathode shows a non-monotonic increasing trend and is consistent with the profile of experimental measurement so that the so-called active zone is formed. In Chapter 4, the power deposition in the low-pressure HCA was studied in simulation. Two main aspects were considered: the power deposition into particles (both electrons and heavy particles) and the power deposition onto the cathode. It was found that the deposited power into particles increases with the rise of discharge current, but there is no effect on the total power deposition onto the cathode. In high-density plasmas, Coulomb collisions between electrons and ions also become very important, especially since a portion of the deposition energy on heavy particles comes mainly from the energy transfer from electrons to ions. It was also found that regardless of external parameters, half of the power deposition onto the cathode always comes from the particle contribution, while the other half is the net contribution of heat transfer and cathode radiation. The HCA model also allows the simulation of multiple discharge modes for low-pressure HCA discharges over a wide range of gas flow rates. It was also shown that the discharge operating conditions and the external magnetic field can change the distribution of the particle flow on the cathode wall. In Chapter 5, the ion sputtering erosion process on the cathode was simulated by coupling the HCA numerical model with the moving grid technique. The results showed that the ion sputtering erosion on the cathode depends on the ion flux and the plasma potential near the cathode wall and that their distribution and magnitude jointly determine the erosion morphology of the cathode. It was also found that the location of the most severe erosion on the cathode is located in the region of the densest ion flux on the cathode wall, rather than in the longitudinal correspondence with the central region of the internal positive column (IPC). The external magnetic fields can mitigate the cathode erosion and reduce the erosion depth, but stronger magnetic fields lead to a concentration of current density at the cathode tip, which can enhance erosion slightly at the cathode outlet end. Finally, the conclusions and innovation highlights were summarized, and prospects for future work were discussed.