This tutorial paper provides a comprehensive overview of decomposition methods for network utility maximization (NUM). It begins by reviewing fundamental concepts such as convexity, Lagrange duality, distributed subgradient methods, and iterative algorithms like Jacobi and Gauss-Seidel. The paper then introduces various decomposition techniques, including primal, dual, indirect, partial, and hierarchical decompositions, focusing on their application to NUM problems. The authors discuss the implications of these decompositions on network architecture and resource allocation, emphasizing the importance of decomposability in achieving efficient distributed solutions. The paper also presents recent examples of systematic searches for alternative decompositions, decoupling techniques for coupled objective functions, and decoupling techniques for coupled constraint sets that are not easily decomposable. Finally, it explores the application of these decompositions to specific problems, such as QoS rate allocation, and compares different distributed algorithms based on their convergence properties and engineering implications.This tutorial paper provides a comprehensive overview of decomposition methods for network utility maximization (NUM). It begins by reviewing fundamental concepts such as convexity, Lagrange duality, distributed subgradient methods, and iterative algorithms like Jacobi and Gauss-Seidel. The paper then introduces various decomposition techniques, including primal, dual, indirect, partial, and hierarchical decompositions, focusing on their application to NUM problems. The authors discuss the implications of these decompositions on network architecture and resource allocation, emphasizing the importance of decomposability in achieving efficient distributed solutions. The paper also presents recent examples of systematic searches for alternative decompositions, decoupling techniques for coupled objective functions, and decoupling techniques for coupled constraint sets that are not easily decomposable. Finally, it explores the application of these decompositions to specific problems, such as QoS rate allocation, and compares different distributed algorithms based on their convergence properties and engineering implications.