This lecture note introduces the rapidly developing concepts of generalized symmetries, focusing on higher-form and higher-group symmetries from the perspectives of high energy physics and condensed matter physics. It discusses the geometric engineering of quantum field theories (QFTs) in string theory and symmetry-protected topological (SPT) phases in condensed matter physics. The note is based on a lecture course given by Yi-Nan Wang and Qing-Rui Wang in February 2023 at Peking University. Section 2 introduces higher-form symmetries, starting with topological operators in ordinary symmetries. It then defines invertible higher-form symmetries, using examples such as Maxwell theory, gauge theories with charged matter, and non-Abelian gauge theories. The section also discusses gauging higher-form symmetries, 't Hooft anomalies, and the three perspectives on higher-form symmetries: topological defect networks, flat connections, and classifying spaces. Section 3 explores 't Hooft anomalies and SPT phases. It discusses the relationship between 't Hooft anomalies and SPT phases, using examples such as the anomalous spin-1/2 at the boundary of a Haldane chain. The section also provides a general classification of 't Hooft anomalies and SPT phases, including higher-form symmetries, fermionic systems, and cobordism. Section 4 discusses applications of higher-form symmetries in string theory and condensed matter physics. It covers geometric engineering of QFTs in string/M-theory, the computation of higher-form symmetries from the topology of extra-dimensional space, and applications in condensed matter physics such as the toric code model and higher-form SPT phases. Section 5 extends the discussion to more general, categorical symmetries, including higher-group symmetries. It introduces basic concepts of category theory, defines strict 2-groups, and discusses their physical interpretations and gauging. The section also briefly describes weak n-groups and non-invertible symmetries. The lecture note provides a comprehensive overview of generalized symmetries, their applications, and their implications in both high energy physics and condensed matter physics.This lecture note introduces the rapidly developing concepts of generalized symmetries, focusing on higher-form and higher-group symmetries from the perspectives of high energy physics and condensed matter physics. It discusses the geometric engineering of quantum field theories (QFTs) in string theory and symmetry-protected topological (SPT) phases in condensed matter physics. The note is based on a lecture course given by Yi-Nan Wang and Qing-Rui Wang in February 2023 at Peking University. Section 2 introduces higher-form symmetries, starting with topological operators in ordinary symmetries. It then defines invertible higher-form symmetries, using examples such as Maxwell theory, gauge theories with charged matter, and non-Abelian gauge theories. The section also discusses gauging higher-form symmetries, 't Hooft anomalies, and the three perspectives on higher-form symmetries: topological defect networks, flat connections, and classifying spaces. Section 3 explores 't Hooft anomalies and SPT phases. It discusses the relationship between 't Hooft anomalies and SPT phases, using examples such as the anomalous spin-1/2 at the boundary of a Haldane chain. The section also provides a general classification of 't Hooft anomalies and SPT phases, including higher-form symmetries, fermionic systems, and cobordism. Section 4 discusses applications of higher-form symmetries in string theory and condensed matter physics. It covers geometric engineering of QFTs in string/M-theory, the computation of higher-form symmetries from the topology of extra-dimensional space, and applications in condensed matter physics such as the toric code model and higher-form SPT phases. Section 5 extends the discussion to more general, categorical symmetries, including higher-group symmetries. It introduces basic concepts of category theory, defines strict 2-groups, and discusses their physical interpretations and gauging. The section also briefly describes weak n-groups and non-invertible symmetries. The lecture note provides a comprehensive overview of generalized symmetries, their applications, and their implications in both high energy physics and condensed matter physics.