Index matrices, developed over more than four decades, provide a novel framework for extending classical matrix calculus to handle otherwise ill-posed operations between matrices of differing dimensions. Originating from early theoretical insights and refined through years of research, they have found applications in generalized nets, graph theory, fuzzy systems, decision-making, and beyond. The book presents the core concepts, key results, and evolving extensions of index matrices, including multidimensional formulations and their associated operations. Through intuitive examples, it demonstrates how seemingly undefined operations can be transformed into mathematically consistent and meaningful ones. The author also highlights some open problems, inviting further exploration in this rich and growing area of research.