Multilayer networks are essential for understanding complex systems that involve multiple types of relationships, dynamic changes, and interconnected subsystems. This paper reviews the history and current state of multilayer network research, discusses various types of multilayer networks, and presents a general framework for analyzing them. It also explores existing data sets, diagnostic tools, and models for multilayer networks, including community structure, connected components, tensor decompositions, and dynamical processes. The paper introduces a general framework for multilayer networks, which allows for the representation of complex systems with multiple layers or aspects. It discusses various types of multilayer networks, such as multiplex networks, interdependent networks, and networks of networks, and provides a classification based on the constraints they impose. The framework is able to handle networks with multiple modes of multiplexity, such as networks that are both multiplex and temporal. The paper also discusses tensor representations of multilayer networks, which allow for the analysis of complex systems using high-dimensional data. It introduces the concept of "supra-adjacency matrices," which are used to represent multilayer networks in a simplified form. These matrices are useful for studying processes such as diffusion, epidemic spreading, and synchronizability on multilayer networks. The paper also discusses the relationship between multilayer networks and other types of networks, such as hypergraphs, multipartite networks, and networks that are both node-colored and edge-colored. It highlights the importance of considering multiple aspects of networks, such as time, interaction type, and node attributes, in the analysis of complex systems. In conclusion, the paper emphasizes the importance of developing a unified framework for multilayer networks to better understand complex systems. It highlights the need for further research into the generalization of single-layer network concepts to multilayer networks and the development of new tools and methods for analyzing these complex systems.Multilayer networks are essential for understanding complex systems that involve multiple types of relationships, dynamic changes, and interconnected subsystems. This paper reviews the history and current state of multilayer network research, discusses various types of multilayer networks, and presents a general framework for analyzing them. It also explores existing data sets, diagnostic tools, and models for multilayer networks, including community structure, connected components, tensor decompositions, and dynamical processes. The paper introduces a general framework for multilayer networks, which allows for the representation of complex systems with multiple layers or aspects. It discusses various types of multilayer networks, such as multiplex networks, interdependent networks, and networks of networks, and provides a classification based on the constraints they impose. The framework is able to handle networks with multiple modes of multiplexity, such as networks that are both multiplex and temporal. The paper also discusses tensor representations of multilayer networks, which allow for the analysis of complex systems using high-dimensional data. It introduces the concept of "supra-adjacency matrices," which are used to represent multilayer networks in a simplified form. These matrices are useful for studying processes such as diffusion, epidemic spreading, and synchronizability on multilayer networks. The paper also discusses the relationship between multilayer networks and other types of networks, such as hypergraphs, multipartite networks, and networks that are both node-colored and edge-colored. It highlights the importance of considering multiple aspects of networks, such as time, interaction type, and node attributes, in the analysis of complex systems. In conclusion, the paper emphasizes the importance of developing a unified framework for multilayer networks to better understand complex systems. It highlights the need for further research into the generalization of single-layer network concepts to multilayer networks and the development of new tools and methods for analyzing these complex systems.