This paper introduces a tensorial framework for studying multi-layer networks, generalizing traditional network theory to handle complex systems with multiple layers of connectivity. The authors propose a mathematical representation using tensors to describe both single-layer (monoplex) and multi-layer networks, including multiplex and temporal networks. They generalize key network descriptors such as degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion processes to the tensorial framework. The tensor approach allows for a systematic comparison between single-layer and multi-layer networks, enabling the analysis of phenomena like multiplexity-induced correlations and new dynamical feedbacks. The framework also facilitates the study of multi-layer networks by considering both intra-layer and inter-layer connections, which are essential for understanding complex systems such as social networks, transportation systems, and biological networks. The authors demonstrate how to compute various network properties using tensor algebra, including the extraction of specific layers, projection of multi-layer networks onto single-layer networks, and the construction of a network of layers to analyze inter-layer relationships. The tensorial approach provides a powerful tool for analyzing and modeling multi-layer complex systems, offering new insights into the structure and dynamics of these systems.This paper introduces a tensorial framework for studying multi-layer networks, generalizing traditional network theory to handle complex systems with multiple layers of connectivity. The authors propose a mathematical representation using tensors to describe both single-layer (monoplex) and multi-layer networks, including multiplex and temporal networks. They generalize key network descriptors such as degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion processes to the tensorial framework. The tensor approach allows for a systematic comparison between single-layer and multi-layer networks, enabling the analysis of phenomena like multiplexity-induced correlations and new dynamical feedbacks. The framework also facilitates the study of multi-layer networks by considering both intra-layer and inter-layer connections, which are essential for understanding complex systems such as social networks, transportation systems, and biological networks. The authors demonstrate how to compute various network properties using tensor algebra, including the extraction of specific layers, projection of multi-layer networks onto single-layer networks, and the construction of a network of layers to analyze inter-layer relationships. The tensorial approach provides a powerful tool for analyzing and modeling multi-layer complex systems, offering new insights into the structure and dynamics of these systems.