Numerical modelling of physical properties and processes in an electric arc stabilized by a water vortex (steam torch) has been summarized in this review paper. One-fluid MHD equations are numerically solved for an axisymmetric thermal plasma flow inside a discharge chamber of the steam plasma torch. The steady state solution results are discussed for the range of currents 300–600 A with relatively low steam flow rate of about 0.3 g s−1. The maximum obtained velocities and temperatures—8500 m s−1, 26,300 K, are reported at the centre of the nozzle exit for 600 A. The evaporation of water, i.e. mass flow rate of steam, was predicted from a comparison between the present simulation and experiments. The generated plasma is mildly compressible (M < 0.7) with the inertial forces overwhelming the magnetic, viscous, centrifugal and Coriolis forces with the factor of 103. Our calculations showed that the most significant processes determining properties of the arc are the balance of the Joule heat with radiation and radial conduction losses from the arc. Rotation of plasma column due to the tangential velocity component has a negligible effect on the overall arc performance, however, the rotation of water induces fluctuations in the arc and in the plasma jet with characteristic frequency which is related to the frequency of rotation of water. Reabsorption of radiation occurs at the radial position higher than 2.5 mm from the arc axis. The amount of reabsorbed radiation is between 17 and 28%. LTE conditions are satisfied in the arc column with the 2 mm radius. Comparison between the present simulations and experiments shows good agreement with the current–voltage characteristics, radial velocity and temperature profiles, as well as with the other related numerical simulation.