Near quenching laminar flames in parallel plate and cylindrical ducts are investigated computationally using one-step chemistry and a two-dimensional finite volume formulation. The effects of varying the heat transfer boundary conditions on the flame shape and propagation speed are examined. Two flame shapes are shown to arise, depending on the channel width and wall heat losses. A quenching criterion is developed for cases of restricted conductive or convective heat loss through the duct walls, and results are compared to the existing theory. As expected, the quenching Peclet number is found to be proportional to the square root of the overall heat loss coefficient. The importance of internal wall radiation and through-wall heat losses to the flame shape and quenching process is also examined and discussed. Radiation inside the channel is shown to inhibit quenching.