This paper is concerned with the capillary problem in a class of non-cylindrical domains in K \subset \mathbb R^{n+1} obtained by scaling a bounded cross-section ­ \Omega \subset \mathbb R^n along the vertical axis. The capillary surfaces are described in two different ways. In the first model, they are described as the boundary of a Caccioppoli set and in a second model, after transforming K to a cylinder, they are described as graphs of functions on \Omega ­. The volume of the fluid is prescribed. For both models, the energy functional is derived and declared on the appropriate function space consisting of BV -functions. Main results are existence and a priori bounds of minimizers, using the direct methods in the calculus of variations. For the special case of a cone over the domain \Omega ­, a criterion is given to assure that the tip is not filled with liquid. Another point of examination concerns modelling the volume restriction by means of a Lagrange multiplier.