This paper is the first of a series in which the theory of local advection (the exchange due to horizontal heterogeneity) of energy and moisture will be developed and applied to a number of problems of practical and theoretical interest. The paper provides an introduction to the practical implications and physical basis of local advection, but it is mainly devoted to developing methods of analysis to be applied in later papers. The treatment aims to provide simple and rapid numerical procedures for the solution of advection problems. Methods are given for solving the two-dimensional atmospheric-diffusion equation subject to “concentration,” “flux,” and “radiation” types of boundary conditions. Appendices give discussions of the properties of the functions entering the solutions and provide simple means for their computation. Extensive tables of the relevant functions are given for the case m = 1/7, n = 6/7 (m, n being the exponents in the power-law approximations to the vertical profiles of mean wind speed and eddy diffusivity, respectively). The rudiments of a quantitative theory of advective inversion are developed, expressions being obtained for the equation of the inversion surface, and for the maximum height and downwind extent of the inversion.