Phonons are quantum mechanical particles corresponding to ionic vibrations. They are similar to electrons in a way that they interact with other particles and defects, and they are responsible for thermal conduction in insulators like electrons are responsible for electrical conduction in conductors. Most of the physical properties due to ionic vibrations can be determined by using harmonic approximation which consider phonons as independent quantum mechanical harmonic oscillators having quadratic potentials depending on the displacements of atoms in their equilbirium positions. However, there are some physical processes such as finite thermal conductivity and thermal expansion which cannot be explained with only harmonic phonons. To investigate these physical processes anharmonicity needs to be taken into account. Anharmonicity is related to the higher order terms in the interatomic potential and corresponds to phonon-phonon interactions. The strength of these interactions depends on the temperature which is related to the available thermal energy, or, the number of phonons given by the Bose-Einstein distribution. In this thesis, the effects of anharmonicity on quantum thermal transport are studied in nanoscale systems by using Green functions. Non-Equilibrium Green Functions (NEGF) method is a perturbative approach to study transport properties of both electronic and phononic systems. Anharmonic terms in interatomic potential are incorporated into NEGF method in the form of a self-energy which can be computed self-consistently. This approach provides high accuracy with high computational cost. As an alternative, mean field technique is computationally more feasible which allows to do calculations for larger systems. In this study, we investigate anharmonic transport properties of one-dimensional chains using NEGF method. Our calculations involve self-energies of third and fourth order anharmonic terms. In addition, mean field calculation for fourth order anharmonicity is performed for comparison.