Molecular dynamics simulations are used to study capillary adhesion from a nanometer scale liquid bridge between two parallel flat solid surfaces. The capillary force, F-cap, and the meniscus shape of the bridge are computed as the separation between the solid surfaces, h, is varied. Macroscopic theory predicts the meniscus shape and the contribution of liquid/vapor interfacial tension to Fcap quite accurately for separations as small as two or three molecular diameters (1-2 nm). However, the total capillary force differs in sign and magnitude from macroscopic theory for h less than or similar to 5 nm (8-10 diameters) because of molecular layering that is not included in macroscopic theory. For these small separations, the pressure tensor in the fluid becomes anisotropic. The components in the plane of the surface vary smoothly and are consistent with theory based on the macroscopic surface tension. Capillary adhesion is affected by only the perpendicular component, which has strong oscillations as the molecular layering changes.