This paper outlines the general solution of the one-dimensional, azimuthally averaged radiative transfer equation in terms of a matrix exponential. The link between this fundamental solution and those more commonly used in radiative transfer is established. The formulation is developed for a general vertically inhomogeneous atmosphere with sources. Several new concepts, based on properties of the matrix exponentials, are described in the context of radiative transfer, including the use of the commutator and product integrals. It is also demonstrated how the matrix exponential formulation provides for new insights, not only into improvements of the numerical efficiency and stability of the solution, but also into the understanding of radiative transfer through a layered atmosphere. The various concepts introduced in this paper are illustrated throughout by the two-stream simplification of the general radiative transfer equation. ¿ American Geophysical Union 1988