Reduced Basis (RB) methods are efficient numerical techniques for solving problems involving repeated solutions of differential equations from engineering and applied sciences. These methods aim to replace high-fidelity approximations with reduced-order models (ROMs) that have significantly smaller dimensions. The core idea is to project the high-fidelity problem onto a subspace spanned by specially selected basis functions, which represent solutions for specific parameters. RB methods have evolved over decades, with foundational work in the 1970s and 1980s, and were further developed in the early 2000s by A.T. Patera and others, leading to significant improvements in computational efficiency through offline-online decomposition and a posteriori error estimation. This textbook provides a comprehensive introduction to RB methods, covering their mathematical formulation, theoretical properties, and practical implementation. It discusses both a priori and a posteriori error analysis, strategies for constructing accurate reduced basis spaces, and offline-online decomposition strategies to reduce computational complexity. The book includes numerous examples of applicative interest, focusing on both linear and nonlinear PDEs. The text is structured into chapters that introduce representative problems, explain the principles of RB methods, and explore their application to various types of differential equations. It also covers extensions to nonlinear and parametrically nonaffine problems using the empirical interpolation method (EIM) and discusses the application of RB methods to optimization and control problems. The book emphasizes the algebraic and geometric structures inherent in RB methods and provides detailed numerical examples to illustrate their effectiveness. The authors acknowledge the contributions of various researchers and collaborators who have influenced the development of RB methods. The book is intended for researchers and students in applied mathematics, engineering, and computational science who are interested in the theory and application of reduced-order modeling techniques. It serves as a valuable resource for understanding the principles and practical implementation of RB methods in solving complex problems efficiently.Reduced Basis (RB) methods are efficient numerical techniques for solving problems involving repeated solutions of differential equations from engineering and applied sciences. These methods aim to replace high-fidelity approximations with reduced-order models (ROMs) that have significantly smaller dimensions. The core idea is to project the high-fidelity problem onto a subspace spanned by specially selected basis functions, which represent solutions for specific parameters. RB methods have evolved over decades, with foundational work in the 1970s and 1980s, and were further developed in the early 2000s by A.T. Patera and others, leading to significant improvements in computational efficiency through offline-online decomposition and a posteriori error estimation. This textbook provides a comprehensive introduction to RB methods, covering their mathematical formulation, theoretical properties, and practical implementation. It discusses both a priori and a posteriori error analysis, strategies for constructing accurate reduced basis spaces, and offline-online decomposition strategies to reduce computational complexity. The book includes numerous examples of applicative interest, focusing on both linear and nonlinear PDEs. The text is structured into chapters that introduce representative problems, explain the principles of RB methods, and explore their application to various types of differential equations. It also covers extensions to nonlinear and parametrically nonaffine problems using the empirical interpolation method (EIM) and discusses the application of RB methods to optimization and control problems. The book emphasizes the algebraic and geometric structures inherent in RB methods and provides detailed numerical examples to illustrate their effectiveness. The authors acknowledge the contributions of various researchers and collaborators who have influenced the development of RB methods. The book is intended for researchers and students in applied mathematics, engineering, and computational science who are interested in the theory and application of reduced-order modeling techniques. It serves as a valuable resource for understanding the principles and practical implementation of RB methods in solving complex problems efficiently.