Nowadays, two-dimensional materials due to their vast engineering and biomedical applications have been the focus of many researches. The present paper proposes a large-deformation theory for thin plates with application to one-atom-thick layers (OATLs). The deformation is formulated exactly in the mathematical framework of Lagrangian description. In particular, an exact finite strain analysis is given - in addition to the usual strain tensor associated to the middle surface, the second and third fundamental forms of the middle surface of the deformed thin plate are also maintained in the analysis. Exact closed-form solutions for a uniaxially curved thin plate due to pure bending in one case and due to a combination of vertical and horizontal loading in another are obtained. As a special case of the latter problem, the exact solution for the plane-strain bulge test of thin plates is derived. Subsequently, the approximation of Vlassak and Nix [Vlassak, J.J., Nix, W.D., 1992. J. Mater. Res., 7(12), 3242-3249] for the load-deflection equation is recovered. The given numerical results are devoted to graphene as the most well-known OATL.