  The Magnus expansion, introduced by Wilhelm Magnus in 1954, is an infinite
Lie series employed to express solutions for first-order homogeneous linear
differential equations involving a linear operator. Since its discovery it has
evolved into a pivotal tool used across diverse disciplines, including physics,
chemistry, and engineering. Over the past 25 years, the Magnus expansion has
undergone significant mathematical developments which revealed an intricate
interplay between algebra, combinatorics, and geometry. By emphasizing a modern
perspectives based on the use of pre- and post-Lie algebras, we discuss the
Magnus expansion from the viewpoint of the notion of crossed morphism.
