Fluid flow models in two spatial dimensions with a one-dimensional interface are known to support overturned traveling solutions. Computational methods of solving the two-dimensional problem are well developed, even in the case of overturned waves. The three-dimensional problem is harder for three prominent reasons. First, some formulations of the two-dimensional problem do not extend to three-dimensions. The technique of conformal mapping is a prime example, as it is very efficient in two dimensions but does not have a three-dimensional equivalent. Second, some three-dimensional models, such as the Transformed Field Expansion method, do not allow for overturned waves. Third, computational time can increase by more than an order of magnitude. For example, the Birkhoff-Rott integral has a cost of O(N2) in two-dimensions but O(N4M2) in three-dimensions, where N is the number of discretized points in the lateral directions and M is the number of truncated summation terms. This study seeks to bridge the gap between efficient two-dimensional numerical solvers and more computationally expensive three-dimensional solvers. The dissertation does so by developing a dimension-breaking continuation method, which is not limited to solving interfacial wave models. The method involves three steps: first, conduct N-dimensional continuation to large amplitude; second, extend the solution trivially to a (N+1)-dimensional solution and solve the linearization; and third, use the linearization to begin (N+1)-dimensional continuation. This method is successfully applied to Kadomtsev-Petviashvili and Akers-Milewski interfacial models and then in a reduced Vortex Sheet interfacial formulation. In doing so, accurate search directions are calculated for use in higher-dimension quasi-Newton solvers.