This article deals with the application of reduced-order models (ROMs), via the asymptotic NNM method, to thin shells large-amplitude vibrations. Two particular geometries are addressed: a doubly-curved shallow shell, simply supported on a rectangular base, and a circular cylindrical panel with simply supported, in-plane free edges. In both cases, the shell is subjected to a harmonic excitation, normal to its surface, and in the spectral neighbourhood of its fundamental frequency. For both shells, the models use Donnell's non-linear strain-displacement relationships, with in-plane inertia retained. The discretized equations of motion are obtained by the Lagrangian approach, where the unknown displacements are expanded on an ad-hoc basis of approximation functions that are not the eigenmodes. As a consequence, a large number of degrees-of-freedom (dofs) is necessary in order to obtain convergence. The reduction to a single NNM is shown for various excitation amplitude, and compared to a reference solution. Perfect results are obtained for vibration amplitude lower or equal to 1.5 times the thickness of the shell.