We have obtained Nth-order response functions for a two-level system described by displaced anharmonic potential surfaces coupled to a heat bath. The anharmonicity of the potentials has been taken into account as a perturbation of harmonic potentials. The heat-bath was assumed to be an ensemble of harmonic oscillators. Coupling between the two-level system and the bath was assumed to be bilinear. The calculations were done analytically using the Liouville-space generating functional, which had been obtained by way of the path-integral approach. The response functions have been defined in terms of line-shape functions with these line-shape functions being expressed in terms of the bath spectral density and the temperature. We have carried out model calculations of the first-, third-, and fifth-order optical processes for various parameters of anharmonicity to show that anharmonicity plays a minor role in linear absorption, impulsive pump-probe, and photon echo experiments, but plays a major role, in some cases, in fifth-order two-dimensional resonant spectroscopy which is proposed in this paper.