We theoretically study the behaviour of vesicles filled with a liquid of higher density than the surrounding medium, a technique frequently used in experiments. In the presence of gravity, these vesicles sink to the bottom of the container, and eventually adhere even on non-attractive substrates. The strong size dependence of the gravitational energy makes large parts of the phase diagram accessible to experiments even for small density differences. For relatively large volume, non-axisymmetric bound shapes are explicitly calculated and shown to be stable. Osmotic deflation of such a vesicle leads back to axisymmetric shapes and, finally, to a collapsed state of the vesicle.