Using the specific functional form D(C)/D-0 = 1+(alphaC)-beta(alphaC)(2) an investigation has been made of (isothermal) transport through a slab membrane under 'simple' boundary conditions and governed by a diffusion coefficient, D(C), which, with increasing concentration, at first increases, passes through a maximum value and finally decreases. The flux, integral diffusion coefficient and concentration profile characteristic of steady-state permeation have been evaluated; special attention has been paid to the positions of such profiles in relation to the corresponding linear distribution associated with a constant diffusion coefficient. The corresponding transient-state transport has been studied within a framework of the time-lag 'early-time' and 'roott' procedures. Expressions for the 'adsorption' and 'desorption' time-lags are given. The concentration-dependence of these time-lags, of the (four) integral diffusion coefficients derived from them and of the arithmetic-mean time-lag ratios have been considered in some detail. The 'early-time' and 'roott' finite-difference procedures have likewise been employed to derive four further integral diffusion coefficients, so enabling a comparison to be made of the nine integral coefficients pertaining to established experimental techniques. Particular interest attaches to the situation for which n =beta(alphaC(0)) = 1 (where C-0 is the ingoing or upstream concentration of diffusant) resulting in D(C-0) being symmetrical about C-0/2. Some consideration has been given, in general, to features of transient-state transport when governed by a symmetrical D(C).