Two models of the wetting dynamics of a non-volatile liquid are introduced. Both models exhibit a transition between partial and complete wetting regimes. The first model, which can also be viewed as an inhomogeneous surface growth model, is studied in a two-dimensional capillary geometry. For complete wetting the profile of the fluid surface, apart from a precursor film, then assumes a stationary shape, which scales with the system size. A precursor film forms even at the transition, which one can locate analytically, although here it moves only as tchi , where chi approximately=1/2. This first model permits only fluctuations parallel to the wall over which the fluid spreads. Monte Carlo simulations of the second model show that the inclusion of fluctuations perpendicular to the wall can lead to qualitatively new phenomena.