Welding, as one of the fundamental and most widespread methods of joining materials in today’s industry, requires constant improvement of existing and development of new execution methods to reduce costs and increase the safety of welded structures. Poor execution results in residual stresses and deformations, which compromise the safety and aesthetics of welded joints. The development of computational techniques for solving numerical problems has led to the creation of various software packages for simulating the welding process. Developed and used to enhance the welding process, these software packages significantly contribute to reducing correction costs on welded structures and optimizing the welding process. Some of the mentioned software packages in this work include Abaqus, SORPAS, Visual Weld, and Simufact Welding. The aim of this work is to analyze different approaches to numerically determining residual stresses. The occurrence of residual stresses, their classification, the processes and factors influencing their appearance in welded joints, measurement methods, and ultimately measures for reducing residual stresses are described. Due to the causal relationship with stresses, the occurrence of residual deformations, existing types, and prevention mechanisms are also explained. Understanding the finite element method, explained in the fourth chapter, is necessary for simulating the welding process. Additionally, the software packages used for simulating the welding process are listed and explained in detail. Finally, three different approaches to modeling the welding process are presented, two of which are conventional, while the third is related to the Abaqus software package.