This research focuses on the calculation of anharmonic vibrational energy levels for polyatomic molecules. The goal is to obtain accurate energy levels for the prediction of spectroscopic properties and thermodynamic quantities, such as entropy and enthalpy. The vibrations for most stable species can be treated as separable and as harmonic oscillators; however, for some molecules, this does not work. Examples of ill-behaved vibrations include low-frequency or "soft" modes, inversion or rotation barriers that are comparable to room temperature, and coupled vibrational modes. The main issues are (1) to find the appropriate ab initio theory level for generating the multidimensional potential energy surface and (2) to explore methods for computing the vibrational energy levels, such as variational, perturbational, and self-consistent field theories.