This paper develops the theory of slow motion of a thin wetting liquid film attached to a meniscus on a solid surface; the liquid is assumed to be completely wetting. The film spreads under Van der Waals forces. To describe the film dynamics, a problem for the evolution equation with boundary conditions at the (unknown) contact line and at the meniscus edge is formulated. The selfsimilar solution is studied. Wetting diffusion kinetics equations are derived. The influence of the curvature radius of an immovable meniscus on the contact line dynamics is described analytically. Wetting is shown to terminate at a certain short radius. A phenomenon of weightlessness of films over a meniscus near a vertical flat wall is demonstrated. Gravitation does not affect the film profile when the film length exceeds the meniscus radius by an order of magnitude. Such an effect is significant only when the film length is much longer.