The Magnus expansion is a powerful tool for solving linear systems of differential equations with varying coefficients, widely used in scientific and engineering fields such as Quantum Mechanics and Control Theory. This review aims to collect and discuss various developments in the Magnus expansion over the past half century, including methods for generating terms, estimates of convergence radius, generalizations, and non-perturbative expansions. It also bridges the Magnus expansion with numerical integration methods, highlighting its applications in solving physical problems. The review covers the historical context, mathematical preliminaries, and detailed analysis of the Magnus expansion, including its connection to time-dependent perturbation theory and graph theoretical analysis. Additionally, it provides illustrative examples and discusses the Magnus expansion's performance compared to perturbative schemes and standard numerical integrators. The review concludes with a comprehensive overview of physical applications, emphasizing the Magnus expansion's utility in solving complex problems in various scientific disciplines.The Magnus expansion is a powerful tool for solving linear systems of differential equations with varying coefficients, widely used in scientific and engineering fields such as Quantum Mechanics and Control Theory. This review aims to collect and discuss various developments in the Magnus expansion over the past half century, including methods for generating terms, estimates of convergence radius, generalizations, and non-perturbative expansions. It also bridges the Magnus expansion with numerical integration methods, highlighting its applications in solving physical problems. The review covers the historical context, mathematical preliminaries, and detailed analysis of the Magnus expansion, including its connection to time-dependent perturbation theory and graph theoretical analysis. Additionally, it provides illustrative examples and discusses the Magnus expansion's performance compared to perturbative schemes and standard numerical integrators. The review concludes with a comprehensive overview of physical applications, emphasizing the Magnus expansion's utility in solving complex problems in various scientific disciplines.