Reduced Basis (RB) methods are a highly efficient approach for numerically approximating problems involving the repeated solution of differential equations, particularly in engineering and applied sciences. This book provides a comprehensive introduction to RB methods, covering their fundamental theoretical properties, algorithmic and implementation aspects, and practical applications. The authors, Alfio Quarteroni, Andrea Manzoni, and Federico Negri, present a general formulation of RB methods, analyze their accuracy and computational efficiency, and discuss strategies for constructing accurate reduced basis spaces. The book also covers the offline/online decomposition strategy to reduce computational complexity and a posteriori error estimation techniques. Additionally, it explores the algebraic and geometric structures of RB methods, the theoretical rationale behind them, and their extension to nonaffine and nonlinear problems. The book includes numerous examples and exercises to illustrate the concepts and methods discussed.Reduced Basis (RB) methods are a highly efficient approach for numerically approximating problems involving the repeated solution of differential equations, particularly in engineering and applied sciences. This book provides a comprehensive introduction to RB methods, covering their fundamental theoretical properties, algorithmic and implementation aspects, and practical applications. The authors, Alfio Quarteroni, Andrea Manzoni, and Federico Negri, present a general formulation of RB methods, analyze their accuracy and computational efficiency, and discuss strategies for constructing accurate reduced basis spaces. The book also covers the offline/online decomposition strategy to reduce computational complexity and a posteriori error estimation techniques. Additionally, it explores the algebraic and geometric structures of RB methods, the theoretical rationale behind them, and their extension to nonaffine and nonlinear problems. The book includes numerous examples and exercises to illustrate the concepts and methods discussed.