This lecture note provides an introduction to the rapidly developing concepts of generalized symmetries, focusing on higher-form and higher group symmetries from both high energy physics and condensed matter physics perspectives. The note is based on a short course given by Yi-Nan Wang and Qing-Rui Wang in February 2023 at Peking University. The content is structured into several sections: 1. **Introduction**: Discusses the fundamental role of symmetry in physics and introduces the two main types of symmetries: global and local (gauge) symmetries. 2. **Higher-Form Symmetry**: Explains the concept of higher-form symmetries, which act on objects extending more than one dimension, such as Wilson loops and 't Hooft loops. It includes examples from Maxwell theory, U(1) gauge theory with charged matter, and non-abelian gauge theories in 4D. 3. **Gauging Higher-Form Symmetry**: Describes the process of gauging higher-form symmetries, including the concept of 't Hooft anomalies and their implications. 4. **Anomalies and SPT**: Introduces 't Hooft anomalies and their relationship with symmetry-protected topological (SPT) phases in condensed matter physics. It covers the classification of SPT phases and higher-form symmetries. 5. **Applications of Higher-Form Symmetry**: Discusses the applications of higher-form symmetries in string theory and condensed matter physics, including geometric engineering of QFTs and the toric code model. 6. **Higher Group Symmetry**: Extends the discussion to more general categorical symmetries, including higher-group symmetries, weak $n$-groups, and non-invertible symmetries. The note aims to provide a comprehensive understanding of generalized symmetries and their applications, making it accessible to both beginners and advanced readers.This lecture note provides an introduction to the rapidly developing concepts of generalized symmetries, focusing on higher-form and higher group symmetries from both high energy physics and condensed matter physics perspectives. The note is based on a short course given by Yi-Nan Wang and Qing-Rui Wang in February 2023 at Peking University. The content is structured into several sections: 1. **Introduction**: Discusses the fundamental role of symmetry in physics and introduces the two main types of symmetries: global and local (gauge) symmetries. 2. **Higher-Form Symmetry**: Explains the concept of higher-form symmetries, which act on objects extending more than one dimension, such as Wilson loops and 't Hooft loops. It includes examples from Maxwell theory, U(1) gauge theory with charged matter, and non-abelian gauge theories in 4D. 3. **Gauging Higher-Form Symmetry**: Describes the process of gauging higher-form symmetries, including the concept of 't Hooft anomalies and their implications. 4. **Anomalies and SPT**: Introduces 't Hooft anomalies and their relationship with symmetry-protected topological (SPT) phases in condensed matter physics. It covers the classification of SPT phases and higher-form symmetries. 5. **Applications of Higher-Form Symmetry**: Discusses the applications of higher-form symmetries in string theory and condensed matter physics, including geometric engineering of QFTs and the toric code model. 6. **Higher Group Symmetry**: Extends the discussion to more general categorical symmetries, including higher-group symmetries, weak $n$-groups, and non-invertible symmetries. The note aims to provide a comprehensive understanding of generalized symmetries and their applications, making it accessible to both beginners and advanced readers.