The main goal of this dissertation is to achieve fundamental understanding of the flow, temperature, and magnetic fields in the whole process of the electroslag remelting (ESR) as well as solidification of the ingot through mathematical modeling and simulation. The main content of this thesis is a combination of nine scientific articles. In addition, a comprehensive review of the mathematical models developed over past decades is given. Experiments, especially those used to verify the numerical models, are also reviewed. Furthermore, features of new technologies originated in the standard ESR process such as ESR with electrode change, ESR with multiple electrodes, current conductive mold (CCM), electroslag rapid remelting (ESRR), pressure-ESR (PESR), and ESR for hollow ingots are discussed. The numerical model, as originally developed by Kharicha and his colleagues [Mater. Sci. Eng. A, 2005, p. 129; Steel Res. Int., 2008, p. 632], was extended in this thesis. With the extended functionalities of the model, it is possible to perform following studies: - The influence of a modeling in two and three dimensions (2D, 3D) on the predicted shape of the melt pool (profile of the solidifying mushy zone of the ingot). - Impacts of electric conductivity of slag (liquid and solid), applied AC frequency, slag cap height, and mold type (either isolated or live) on the electric current path, and their influences on the flow field, temperature field, and solidification of the ingot. - Effects of the crystal morphological parameters such as permeability and primary dendrite arm space (PDAS) on the predicted pool profile of ingot. - The effect of movement of slag-pool interface on the overall electrical resistance and subsequently generated power in the process. - The influences of power interruption during electrode change procedure on the flow and temperature fields as well as solidification of the ingot. - Effects of physicochemical properties of the slag such as thermal and electrical conductivities on the melt rate, shape, and immersion depth of an ESR electrode. Some important knowledge was obtained: - The velocity field in the slag and bulk of melt pool is transient and in 3D feature, but the pool profile of the steel ingot is firmly steady and axisymmetric for an industrial scale ESR process. - The pool profile of the ingot is very sensitive to the interdendritic melt flow in the mushy zone although the velocity magnitude of the interdendritic velocity is notably smaller than the velocity in the bulk of melt pool or slag. - The electric current path governs the velocity and temperature fields as well as pool profile of the ingot, and the possible current path through the mold is not ignorable. - No significant change in the pool profile of the ingot was predicted during the short time of power off (<5 min) through electrode change. - The melt rate, shape, and immersion depth of an ESR electrode depend strongly on the physicochemical properties of the slag. The ratio of the melt rate to the generated power (power consumption) determines the shape of electrode tip. Finally, important directions for further research are pointed out.