This paper addresses the issue of determining the joint shape formed after molten metal re-solidification at the peak brazing temperature. A theoretical approach for modelling two- and three-dimensional joint shapes in situations where the mating surfaces are neither plane nor orthogonal is presented. The approach is based on a variational principle involving the minimization of the potential energy of the molten metal liquid just prior to the onset of solidification. The numerical solution of the variational problem is obtained by employing a parametric finite element method in conjunction with a direct optimization algorithm. The results from the theory are verified by comparison with experimental data obtained from a set of controlled atmosphere brazing tests of aluminium alloys. The results obtained from the theory are in good agreement with the experimental data and empirical evidence. An analysis of the influence of geometry, configuration and orientation of mating surfaces on both two- and three-dimensional brazed joint shapes is presented in an accompanying paper.