This tutorial paper provides an overview of decomposition methods for network utility maximization (NUM), focusing on the role of decomposability in distributed resource allocation and network design. It reviews key concepts such as convex optimization, Lagrange duality, and distributed subgradient methods, and introduces various decomposition techniques including primal, dual, indirect, partial, and hierarchical decompositions. The paper discusses how these methods can be applied to solve NUM problems, emphasizing the importance of decomposability in enabling distributed algorithms that converge to global optima. It also presents recent examples of systematic search for alternative decompositions, decoupling techniques for coupled objective functions, and decoupling techniques for coupled constraint sets. The paper highlights the importance of decomposability in network design, as it allows for modular and distributed control of networks, and discusses the implications of different decomposition approaches on network architecture and performance. The paper concludes with a discussion of the implications of decomposition methods for network utility maximization, emphasizing the need for a systematic understanding of decomposability structures to design efficient and scalable distributed algorithms.This tutorial paper provides an overview of decomposition methods for network utility maximization (NUM), focusing on the role of decomposability in distributed resource allocation and network design. It reviews key concepts such as convex optimization, Lagrange duality, and distributed subgradient methods, and introduces various decomposition techniques including primal, dual, indirect, partial, and hierarchical decompositions. The paper discusses how these methods can be applied to solve NUM problems, emphasizing the importance of decomposability in enabling distributed algorithms that converge to global optima. It also presents recent examples of systematic search for alternative decompositions, decoupling techniques for coupled objective functions, and decoupling techniques for coupled constraint sets. The paper highlights the importance of decomposability in network design, as it allows for modular and distributed control of networks, and discusses the implications of different decomposition approaches on network architecture and performance. The paper concludes with a discussion of the implications of decomposition methods for network utility maximization, emphasizing the need for a systematic understanding of decomposability structures to design efficient and scalable distributed algorithms.