This paper provides an introduction to the theory of algebraic and topological quantum groups, inspired by S. L. Woronowicz. It covers fundamental concepts in Hopf algebra theory, compact quantum groups, and their actions on quantum spaces. The text explains the basic properties of quantum groups, including the classification of quantum SL(2)-groups, their real forms, and quantum spheres. It also discusses quantum Lorentz groups and explores the physical motivations for quantum groups, such as their role in integrable models, conformal field theory, and quantum space-time models. The paper outlines the mathematical structures underlying quantum groups, including the definitions of polynomials on classical matrix groups, corepresentations, and the representation theory of quantum groups. It also presents key examples, such as the quantum SL(2)-groups, and discusses the properties of their matrix elements and the relationships between different quantum group structures. The text emphasizes the importance of quantum groups in both mathematics and physics, particularly in understanding the behavior of quantum systems and the structure of quantum spaces.This paper provides an introduction to the theory of algebraic and topological quantum groups, inspired by S. L. Woronowicz. It covers fundamental concepts in Hopf algebra theory, compact quantum groups, and their actions on quantum spaces. The text explains the basic properties of quantum groups, including the classification of quantum SL(2)-groups, their real forms, and quantum spheres. It also discusses quantum Lorentz groups and explores the physical motivations for quantum groups, such as their role in integrable models, conformal field theory, and quantum space-time models. The paper outlines the mathematical structures underlying quantum groups, including the definitions of polynomials on classical matrix groups, corepresentations, and the representation theory of quantum groups. It also presents key examples, such as the quantum SL(2)-groups, and discusses the properties of their matrix elements and the relationships between different quantum group structures. The text emphasizes the importance of quantum groups in both mathematics and physics, particularly in understanding the behavior of quantum systems and the structure of quantum spaces.