We describe flame propagation in a channel subject to a Poiseuille flow within the context of a thermo-diffusive model under (i) adiabatic conditions, and (ii) in the presence of conductive heat losses at the walls. The curved two-dimensional lfames examined may be either assisted or opposed by the flow. For the adiabatic case, we focus attention on the two parameters that characterize the flow field: the mean flow rate and the channel width. The total burning rate and the propagation speed are determined in terms of these parameters. The results may be useful in interpreting features encountered in more complicated flows, which are also characterized by the lfow intensity and a spatial scale. For the non-adiabatic case, we focus attention on the intensity of heat loss. We first consider the case of no external flow. Two modes of extinction are uncovered: global extinction where flame propagation is not possible, and partial extinction where extinction occurs only near the walls with the flame surviving at the center of the channel. Next, we examine the effect of the flow and show that a flow directed from the unburned towards the burned gas makes the flame more vulnerable to heat losses while a flow directed towards the unburned mixture makes it less sensitive.