Axial magnetic field（AMF） technology is a most important control method of vacuum arc（VA）.Numerical simulation is an important tool for the research of the physics mechanisms of vacuum arcs as it is more efficient and economical. In this paper

a review of the research progress of simulations of the vacuum arc under AMF is presented

from early models to two-dimensional magneto-hydrodynamic models

then to three-dimensional ones under the actual AMF distribution. The influence of the anode vaper is taken into account based on the three-dimensional model. The main weakness of present high current vacuum arc（HCVA）models is analyzed. We consider it is mainly due to the insufficient consideration of the facts that the arc is fundamentally a dynamic object. The magnetic field distribution has significant influence on the characteristics of vacuum arc. The expectation of developments of the modeling and simulation vacuum arcs in the future is given. Models and simulations of vacuum arc will be developed toward more complete direction

including interaction between arc column

cathode spots and anode activity. In the future

the 3D vacuum arc model should be developed and multi-component vacuum arc should be taken into account.