This thesis describes the development and testing of a new suite of methods for solving the nuclear vibrational Schrödinger equation in order to calculate anharmonic fundamental vibrational frequencies for realistically sized chemical systems. To get around the potential energy surface (PES) construction bottleneck, we present a new method based upon constructing the PES in a curvilinear coordinate system and transforming back to rectilinear normal mode coordinates to facilitate solving the vibrational problem. We also implement and benchmark the performance of a screened vibrational configuration interaction method for calculating anharmonic fundamental frequencies. Both methods combined allow modelling of vibrational spectra for molecules with up to 20 atoms to be calculated routinely on a desktop computer, provided that the ab initio calculations required to construct the PES are computationally feasible.