Heat transfer by conduction in the walls of mini-micro-channels can get a quite multidimensional character. The wall heat flux density, for small Reynolds numbers, can become strongly non-uniform: most of the flux is transferred to the fluid flow at the entrance of the mini-micro-channel. Two analytical models of channel between parallel plates are proposed. It is shown that inverting experimental measurements using a one-dimensional model, for small flow rates, may lead to an underestimation of heat transfer coefficients. Another conclusion is that axial conduction in the walls of a mini-micro counter-flow heat exchanger yields a loss of efficiency: an optimal wall conductivity that maximises this efficiency exists. A new non-dimensional number quantifying the part of axial conduction in walls is proposed. (C) 2004 Elsevier Ltd. All rights reserved.