We call polynomial approximation a particular form of perturbation theory based on the expansion of a nonpolynomial potential-energy function in a Taylor series about a conveniently chosen coordinate point, in a way similar to the approach known as small-amplitude oscillation in classical mechanics. This approximate method commonly gives more accurate results for deep wells and energies close to the minimum of the potential-energy function. Throughout this chapter we discuss several polynomial approximations, including the celebrated large-N expansion and its variants [148].