A very simple Lagrangian finite difference scheme has been developed to calculate the time dependent advection of air pollutants. It is mass conserving and avoids numerical pseudo-diffusion. No condition of numerical stability is required. The Eulerian grid used for the diffusion part of the pollutant transport equation remains unchanged. There are no restrictions on temporally and spatially variable emission rates, production and destruction processes, wind velocity, diffusion coefficients, roughness parameters or inversion heights. The only exception is that the wind field should not be too far from being homogeneous in the horizontal direction (test of D. W. Pepper and P. E. Long, 1978, J. appl. Met. 17, 228–233). Steady state solutions are nearly identical with corresponding analytical solutions. The propagation of a pollutant cloud is simulated more realistically as compared with the advection treatment of E. Runca and F. Sardei (1975) Atmospheric Environment 9,69–80) and M. Dunst (1980 Z. Met. 30, 47–59). The course of a diffusion experiment is modelled to demonstrate the efficiency of the proposed method. Because of its simplicity, the method is especially suited for use in license processes, for control and for calculating health risks in relation to industrial and power plant accidents with the goal of organizing efficient protection or evacuation.