This book, "Tame Algebras and Integral Quadratic Forms," edited by A. Dold and B. Eckmann and authored by Claus Michael Ringel, is a comprehensive study of the representation theory of finite-dimensional algebras, with a focus on tubular algebras. The book aims to provide an introduction to the new representation theory developed over the past 15 years and to explore the structure of module categories of a specific class of algebras called tubular algebras. The first chapter delves into the study of integral quadratic forms, which are crucial for understanding the representation theory of finite-dimensional algebras. It includes detailed proofs of theorems by Ovsienko and classifications of graphical forms. The second chapter introduces quivers, module categories, and subspace categories, providing foundational concepts necessary for the study of tubular algebras. Chapters 3 and 4 focus on the construction of stable separating tubular families and the properties of tilting modules, respectively. These chapters are essential for understanding the structure of tubular algebras and their module categories. Chapter 5 provides a detailed study of the module categories of tubular algebras, including the structure theorem for these categories and the role of tubular families. Chapter 6 deals with directed algebras, classifying large sincere directed algebras and providing bounds on the length of indecomposable modules. The book also includes an appendix with a complete list of all tame concealed algebras and a discussion of tubular vectorspace categories. Throughout the text, the author emphasizes the combinatorial approach to representation theory, using concepts like Auslander-Reiten quivers and tube families to analyze the structure of module categories. The book concludes with a discussion of directed algebras and their classification. Overall, the book serves as a valuable resource for researchers and students interested in the representation theory of finite-dimensional algebras, particularly those working on tubular algebras and their module categories.This book, "Tame Algebras and Integral Quadratic Forms," edited by A. Dold and B. Eckmann and authored by Claus Michael Ringel, is a comprehensive study of the representation theory of finite-dimensional algebras, with a focus on tubular algebras. The book aims to provide an introduction to the new representation theory developed over the past 15 years and to explore the structure of module categories of a specific class of algebras called tubular algebras. The first chapter delves into the study of integral quadratic forms, which are crucial for understanding the representation theory of finite-dimensional algebras. It includes detailed proofs of theorems by Ovsienko and classifications of graphical forms. The second chapter introduces quivers, module categories, and subspace categories, providing foundational concepts necessary for the study of tubular algebras. Chapters 3 and 4 focus on the construction of stable separating tubular families and the properties of tilting modules, respectively. These chapters are essential for understanding the structure of tubular algebras and their module categories. Chapter 5 provides a detailed study of the module categories of tubular algebras, including the structure theorem for these categories and the role of tubular families. Chapter 6 deals with directed algebras, classifying large sincere directed algebras and providing bounds on the length of indecomposable modules. The book also includes an appendix with a complete list of all tame concealed algebras and a discussion of tubular vectorspace categories. Throughout the text, the author emphasizes the combinatorial approach to representation theory, using concepts like Auslander-Reiten quivers and tube families to analyze the structure of module categories. The book concludes with a discussion of directed algebras and their classification. Overall, the book serves as a valuable resource for researchers and students interested in the representation theory of finite-dimensional algebras, particularly those working on tubular algebras and their module categories.