This textbook is an introduction to the techniques of group theory and covers all important concepts in this field, with a focus on finite groups. It starts where group theory begins: with permutation groups. It then addresses essential structures and methods, such as working with commutators and constructing new groups from given ones. The next goal is the Fitting group and its generalization, which involves studying nilpotent groups. Following that, the text turns to simple groups. Finally, the simplicity of projective linear groups is proven, and an overview of orthogonal, symplectic, and unitary groups is provided. Additionally, the sporadic Mathieu groups and the Higman-Sims group are constructed. The book is written for students in Bachelor's and Master's programs.