A capillary based adhesion device motivates the study of coupled free-interface shapes and the transition from the drop to bridge shape. When a large number of drops, pinned at circular contact lines, are touched to a surface they form liquid bridges, and these bridges create an adhesive force. Alternatively, if the drops are not brought to the surface quickly enough the drops will coarsen, forming instead one large drop. Consider first the coarsening process. The dissipation occurs primarily in the conduits, the drop retain their equilibrium shape - the spherical cap. Drops scavenge volume from one another based on pressure differences, proportional to the surface tension, and arising from curvature differences. This process minimizes the total surface energy. All fixed points and their linear stabilities, obtained analytically, are found to be independent of connectivity. The system coarsens in the sense that, with time, volume is increasingly localized and ends up in a single 'winner' drop. To determine which of the stable fixed points will be the winner, manifolds separating the attracting regions are found using a method which combines local information (eigenvectors at fixed points) with global information (invariant manifolds due to symmetry). The coarsening rate is predicted heuristically, with the Lifshitz-Slyozov-Wagner (LSW) model and compared against numerical simulations for a variety of networks. Distributions of large drop volumes from LSW are independent of network topology; in contrast, simulation results depend weakly on the network dimension. When a pinned drop touches a solid surface it forms a liquid bridge; here the energy is dissipated within the bridge. The dissipated energy is equal to the loss of surface energy, which can also be expressed in terms of forces along the interface using a geometric relation. This energy balance provides an extra relation which determines the microscopic nature of the contact line. Boundary integral method simulations are used to compute the flow field and viscous bending of the free interface. The energy balance is applied to simulations to find slip lengths. The energy balance is used to bound the microscopic contact angle analytically.