During the last 35 years, well over 100 algorithms for modeling advection processes have been described and tested. This review summarizes the development and improvements that have taken place. The nature of the errors caused by numerical approximation to the advection equation are highlighted. Then the particular devices that have been proposed to remedy these errors are discussed. The extensive literature comparing transport algorithms is reviewed. Although there is no clear cut “best” algorithm, several conclusions can be made. The judicious use of simple finite difference schemes (second‐order time differences and even‐order (>2) spatial differences) provides a minimum level of accuracy that is suitable for many atmospheric applications. More complex schemes can yield a significant improvement in accuracy, but sometimes at great computational expense. Spectral and pseudospectral techniques consistently provide the highest degree of accuracy, but expense and difficulties assuring positive mixing ratios are serious drawbacks. Schemes which consider fluid slabs bounded by grid points (volume schemes), rather than the simple specification of constituent values at the grid points, provide accurate positive definite results. The computer memory requirements of the volume schemes can be excessive. Recent attempts to maxmize accuracy while keeping cost low have lead to such useful schemes as the one proposed by P. K. Smolar