Chemical Organization Theory (COT) provides a structural and algebraic framework for analyzing reaction-based systems independently of kinetic assumptions. By identifying closed and self-maintaining sets of species, called organizations, the theory characterizes all structurally admissible persistent configurations of a reaction network. This review synthesizes the mathematical foundations of COT, including closure operators, stoichiometric feasibility, and lattice structures, together with their algorithmic and dynamical interpretations. We examine how organizational structure constrains long-term behavior in reaction-based dynamical systems, including ordinary differential equations, stochastic processes, and spatial reaction-diffusion models. Computational methods for enumerating organizations and distributed organizations are reviewed, alongside extensions to discrete and stochastic settings. Applications spanning atmospheric chemistry, virus dynamics, gene regulation, cell cycle models, and data-driven reaction systems illustrate the breadth and versatility of the framework. Relations to chemical reaction network theory, autocatalytic set theory, and constraint-based approaches are clarified to position COT within the broader landscape of mathematical reaction network analysis. We conclude by highlighting open problems related to transient dynamics, structural transitions, computational scalability, and evolutionary processes, and by emphasizing COT as a unifying structural abstraction for persistence and qualitative behavior in complex reaction systems.