The chapter introduces the concept of residuals and their role in regression analysis, emphasizing their importance in diagnosing model assumptions and identifying outliers. It begins with a detailed explanation of the hat matrix, which projects the response vector onto the column space of the design matrix, and discusses the properties of the diagonal elements of the hat matrix, known as leverage values. The chapter then explores various types of residuals, including ordinary residuals, studentized residuals, and residuals with specific covariance structures. It highlights the advantages of studentized residuals in handling scale dependence and their application in detecting outliers. The chapter also provides examples and illustrations to demonstrate how these residuals can be used to assess the appropriateness of the regression model and to identify influential observations. Finally, it discusses the limitations of using residuals alone and the importance of combining them with other diagnostic tools for a comprehensive analysis.The chapter introduces the concept of residuals and their role in regression analysis, emphasizing their importance in diagnosing model assumptions and identifying outliers. It begins with a detailed explanation of the hat matrix, which projects the response vector onto the column space of the design matrix, and discusses the properties of the diagonal elements of the hat matrix, known as leverage values. The chapter then explores various types of residuals, including ordinary residuals, studentized residuals, and residuals with specific covariance structures. It highlights the advantages of studentized residuals in handling scale dependence and their application in detecting outliers. The chapter also provides examples and illustrations to demonstrate how these residuals can be used to assess the appropriateness of the regression model and to identify influential observations. Finally, it discusses the limitations of using residuals alone and the importance of combining them with other diagnostic tools for a comprehensive analysis.