The dynamics and stability of a thin, viscous film of volatile liquid flowing under the influence of gravity over a non-uniformly heated substrate are investigated using lubrication theory. Attention is focused on the regime in which evaporation balances the flow due to gravity. The film terminates above the heater at an apparent contact line, with a microscopically thin precursor film adsorbed due to the disjoining pressure. The film develops a weak thermocapillary ridge due to the Marangoni stress at the upstream edge of the heated region. As for spreading films, a more significant ridge is formed near the apparent contact line. For weak Marangoni effects, the film evolves to a steady profile. For stronger Marangoni effects, the film evolves to a time-periodic state. Results of a linear stability analysis reveal that the steady film is unstable to transverse perturbations above a critical value of the Marangoni parameter, leading to finger formation at the contact line. The streamwise extent of the fingers is limited by evaporation. The time-periodic profiles are always unstable, leading to the formation of periodically-oscillating fingers. For rectangular heaters, the film profiles after instability onset are consistent with images from published experimental studies. (C) 2010 Elsevier Inc. All rights reserved.