In my PhD dissertation, I studied stable capillary surfaces with planar or spherical boundary in the absence of gravity. I will introduce both problems and our advances towards them. In the case where the boundary lies in a plane we show that the only immersed stable capillary surfaces with embedded boundary are the spherical caps. When the capillary surface lies inside the euclidean unit ball with its boundary on the unit sphere, we construct a Killing vector field for the hyperbolic metric and use it to show that if the center of mass of the region bounded between the surface and the unit sphere is at the origin, the configuration cannot be stable. As a corollary of this approach we obtain a new proof of a theorem by Barbosa and Do Carmo.