This chapter is concerned with the matter of mathematically modelling and computationally simulating the thermo and fluid dynamical phenomena occuring in the workpiece during a gas metal arc welding (GMAW) process, and does so by employing a continuum mechanical approach and a finite element formulation for approximating the solution of equations expressing the continuity of mass, the balance of linear momentum, the conservation of energy and the motion of the weld pool surface. GMAW is an electrode arc fusion welding process. The designation arc fusion signifies that an electric arc is struck between the welding electrode and the workpiece, and this causes the base material to melt on either side of the joint. During the subsequent solidification this will cause fusion between the workpiece parts. The electrode consist in a filler metal, and it is hence consumed during the process and molten droplets are, under the influence of electromagnetical and gravitational forces, transferred to the liquid weld pool. Mass is thus added to the workpiece and this causes a reinforcement of the joint.