The paper introduces a tensorial framework to study multi-layer networks, which are essential for understanding complex systems with multiple subsystems and layers of connectivity. Traditional network theory, which primarily uses adjacency matrices for single-layer networks, is insufficient for analyzing multiplex and time-dependent networks. The authors develop a more general mathematical framework to handle multi-layer networks, including multiplex networks, temporal networks, and interdependent networks. They define and quantify various network descriptors and dynamical processes, such as degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion, within this framework. The paper also discusses the impact of different choices in constructing these generalizations and provides examples of how to obtain known results for single-layer and multiplex networks. The tensorial approach is useful for addressing challenges in multi-layer complex systems, such as inferring influence relationships in multichannel social networks and developing routing techniques for multimodal transportation systems.The paper introduces a tensorial framework to study multi-layer networks, which are essential for understanding complex systems with multiple subsystems and layers of connectivity. Traditional network theory, which primarily uses adjacency matrices for single-layer networks, is insufficient for analyzing multiplex and time-dependent networks. The authors develop a more general mathematical framework to handle multi-layer networks, including multiplex networks, temporal networks, and interdependent networks. They define and quantify various network descriptors and dynamical processes, such as degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion, within this framework. The paper also discusses the impact of different choices in constructing these generalizations and provides examples of how to obtain known results for single-layer and multiplex networks. The tensorial approach is useful for addressing challenges in multi-layer complex systems, such as inferring influence relationships in multichannel social networks and developing routing techniques for multimodal transportation systems.