A brief review of the evolution of the diffusion boundary layer (DBL) conception inspired by the works of Nernst, Levich and Amatore is presented. Experimental methods for studying the DBL in electrode and membrane systems are considered. The electrochemical behaviour of a CM2 cation-exchange membrane in NaCl and KCl solutions is studied by chronopotentiometry at constant under-limiting current. Chronopotentiometric curves are described theoretically by applying the Kedem-Katchalsky equations in differential form to a three-layer system including the membrane and two adjoining DBLs. The conductance coefficients entering the equations are found by treating the results of membrane characterisation: the electrical conductivity, transport numbers of ions and water, electrolyte uptake, as functions of the equilibrium electrolyte solution. The two-phase microheterogeneous model is used for this treatment resulting in presentation of the conductance coefficients as functions of (virtual) electrolyte solution concentration in the membrane. The steady-state DBL thickness (delta) is found by fitting experimental potential drop at sufficiently high times. It is found that delta is proportional to (Delta c)(-0.2), where Delta c is the difference between the electrolyte concentration in the solution bulk and at the interface. This result differs from the Levich equation, which gives the power equal to -0.25 for Delta c. This deviation is explained by the fact that the theory of Levich does not take into account microscopic chaotic convection motion recently described by Amatore et al. It is shown that the treatment of experimental chronopotentiometric curves with the model developed allows one to observe the role of streaming potential in the membrane. Different mechanisms of streaming potential and their effect on the shape of chronopotentiograms are discussed. A simple analytical solution of Navier-Stokes equations applied to natural convection near an infinite vertical ion-exchange membrane is found. It is shown that the formation of DBL induced by electric current is quasi-stationary. This fact allows the empirical expression found earlier and linking delta with Delta c under steady-state conditions to be used in transient regimes. The numerical solution of the non-stationary Kedem-Katchalsky equations together with this empirical expression results in quantitative description of the potential difference (pd) and delta as functions of time in chronopotentiometric experiments. The comparison of theoretical and experimental chronopotentiometric curves shows an excellent agreement, especially for the part after switching off the current. The reasons of a small deviation observed just before the curves attain steady state under a constant current applied are discussed.