A comprehensive, unified approach to radiative transfer in astronomical maser sources is presented. A general formalism and detailed analytic solutions are given for the linear maser, including the case of a source illuminated by background radiation. A new, general expression is presented for the product of intensities of the two streams on any given ray. The complete analytic solution of a linear maser without background radiation is developed, and the results of the numerical calculations of Alcock and Ross (1985) are explained. A detailed solution is presented of a three-dimensional maser of an arbitrary shape. This general solution is then expressed in terms of the escape probability approach, providing for the first time a rigorous, formal foundation for the employment of this useful method in numerical solutions of the level populations of a maser system. The general solution is applied to the spherical geometry, discussing the observational implications.