Until present, many researchers relied on the conventional plug flow dispersion models to analyse the concentration profiles obtained from the tracer injection experiments to evaluate the dispersion coefficients in packed beds. The Fickian concept in the limit of long time duration is assumed to be applicable and it implies that the mean-square displacement of the tracer profile is constant with time and the concentration profile is Gaussian. There were very few studies on identifying the conditions under which this assumption is valid and delineate the range of applicability of the existing plug flow dispersion models. If the time scales of a tracer injection experiment are not sufficient for a tracer to traverse the bed radius and sample the velocity variations, this could give rise to persisting non-Fickian transients where the mean-square displacement of the tracer profile is not constant with time and the concentration profile deviates from the normal Gaussian distribution. These transients cannot be predicted by the conventional plug dispersion models. An extended axial non-Fickian macroscopic dispersion model is derived to describe the transient development of a solute tracer when injected into a fluid flowing through a cylindrical packed bed or empty tube and some non-Fickian effects in the dispersion process. The flow profile in beds packed with uniform particles exhibits radial non-uniformity due to the oscillatory variation in porosity because of the wall confinement (wall effect). Compared with the axial plug flow dispersion model, the extended model contains time-dependent coefficients such as the transient axial dispersion coefficient and higher order derivatives (higher than second order) of the cross-sectionally averaged concentration. Including them provides some insight on non-Fickian transport in the dispersion process. The model provides time criteria on the basis that the effelongitudinal dispersion coefficient in the packed bed reaches its asymptotic value and the non-Fickian transients will die out. Some experimental conditions in the literature were checked by these criteria and found to be either marginally satisfied, or not satisfied at all, which indicates that the Fickian concept is not valid. The model results for tracer dispersion in cylindrical packed beds show that the longitudinal dispersion coefficient converges to its asymptotic value on a time scale proportional to R2/(DT) where R is the column radius and (DT) is the area averaged lateral dispersion coefficient. The extended model encouraged study of the consequences of the additional dispersion terms in other applications such as the pulse spread in the field of capillary column inverse gas chromatography (CCIGC). CCIGC is used to evaluate the solute-polymer diffusion coefficient Dp and the partition coefficient K at infinite dilute conditions. The tube geometry in CCIGC is more complex than the conventional Taylor dispersion problem due to the polymer coating on the inside of the capillary wall. The extended CCIGC model presented in this study has advantages over the previous models by including the effects of Taylor dispersion and higher order derivatives of the pulse area-averaged concentration. Taylor dispersion effect causes more pulse spread in the longitudinal direction and by not including it in the CCIGC regression models may cause a significant error in the measured Dp values. The extended CCIGC model provides for the first time criteria on capillary dimensions for the transient coefficients (multiplying the second and higher order derivatives) to become constant and for the non-Fickian effects associated with the higher order derivatives to be neglected. Model results show that Taylor dispersion effect has a significant effect on the elution profiles at high values of Dp and/or low values of gas diffusion coefficients Dg and it can be used to increase the sensitivity range of the previous CCIGC models at extremely low and high Dp values.