Thin elastic sheets are found throughout nature and are also extremely important in industrial applications. Sheets can be described using plate and shell models which, in systems where shear is negligible, are typically fourth-order, two-dimensional, partial differential equations. Many such models exist; however, the circumstances in which a particular model is appropriate to use may not be readily apparent. Therefore a means of comparing different unshearable plate and shell models in a general setting is of interest, and is yet to be systematically addressed in the literature. The focus of this thesis is the implementation of a generic numerical framework for the discretization of two-dimensional, fourth-order boundary-value problems, using the method of boundary patches. We build upon the literature for curved triangular Hermite elements by outlining the explicit construction formulas for a known class of curved elements, compatible with Bell elements. We implement these elements within the finite element library oomph-lib . In this study we consider three plate models: the well-known moderate-rotation Foppl-von Karman model, the arbitrary-rotation Koiter-Steigmann plate model and a new moderate- to-large rotation model, which we derive herein. We then implement these plate models within the library, so that they can be solved on generic domains. Finally, we use the implemented plate models, along with analytic and finite difference approaches, to compare the models in three different contexts. The systems we study are the clamped inflation of a circular sheet, the inflation of a circular sheet subject to a rolling clamp, which undergoes a wrinkling instability, and the large cantilever-type displacement of a complicated curved domain, respectively. In all of these systems the choice of plate models is demonstrated to be important: in particular in all cases the predictions of the Foppl-von Karman model break down for moderate-thickness sheets, yielding inaccurate predictions of the sheet morphology. This serves both to demonstrate the capability of the numerical framework as a comparison tool and highlight why such comparisons are important.