On graph approaches to contextuality and their role in quantum theory Chapter 01 Introduction Chapter 02 Contextuality: the Compatibility Hypergraph Approach Chapter 03 Contextuality: the Exclusivity Graph Approach Chapter 04 The Exclusivity Principle and Its Consequences Appendix A State independent proofs of the Bell Kochen Specker Theorem This book explores two of the most striking features of quantum theory contextuality and nonlocality using a formulation based on graph theory. Quantum theory provides a set of rules to predict probabilities of different outcomes in different experimental settings and both contextuality and nonlocality play a fundamental role in interpreting the outcomes. In this work the authors highlight how the graph approach can lead to a better understanding of this theory and its applications. After presenting basic definitions and explaining the non contextuality hypothesis the book describes contextuality scenarios using compatibility hypergraphs. It then introduces the exclusivity graph approach which relates a number of important graph theoretical concepts to contextuality. It also presents open problems such as the so called Exclusivity Principle as well as a selection of important topics like sheaf theoretical approach hypergraph approach and alternative proofs of contextuality