This thesis presents research carried out as part of the ExoMol project, towards calculating theoretical spectra for the main isotopologue of acetylene, 12C2H2, for use in characterising hot exoplanet or cool stellar atmospheres. A large component of this work was in the development of numerical methods for treating linear polyatomic molecules such that these calculations could be carried out in an efficient and feasible way; ro-vibrational calculations of linear molecules are very non-trivial and require a unique treatment in order to avoid singularities in the Hamiltonian. A novel approach was employed in variational nuclear motion programme TROVE, which involves the use of a finite Dnh symmetry group and classication of ro-vibrational states using the vibrational angular momentum operator, Lz. This has been used in nuclear-motion calculations to compute an ab initio linelist of 12C2H2 covering 13.9 million transitions between 2.7 million states, up to a rotational excitation of J = 58. In order to facilitate an accurate calculated spectra, available experimental data of 12C2H2 were collated and analysed to obtain an accurate set of 11,213 empirical energy levels using the MARVEL procedure. As demonstrated, these can be used to produce a high-accuracy potential energy surface and subsequent semi-empirical model for the ro-vibrational energies and intensities of acetylene, which can be computed up to high ro-vibrational excitations. Calculations using this semi-empirical model are in progress for an accurate high-temperature linelist, expected to be valid up to 1000-1200 K. This will be published in due course and will be appropriate for characterising exoplanet and cool stellar atmospheres; these ongoing calculations are discussed.