Abstract: The Magnus expansion was first introduced by Wilhelm Magnus in his 1954 paper, “On the Exponential Solution of Differential Equations for a Linear Operator” (CPAM 7 (1954) 649), where he addressed a key problem in applied mathematics: the computation of the logarithm of the operator- or matrix-valued solution to a linear initial value problem. Since its discovery, the Magnus expansion has evolved into a useful tool, used across various fields such as physics, chemistry, and engineering. Over the past 25 years, mathematical developments have revealed connections between algebra, combinatorics, and geometry in the context of the Magnus expansion. These advances have clarified its structure and extended its applications.