An expression for the total, elastic and inelastic, differential cross section for scattering of thermal neutrons by a rigid polyatomic molecule is derived. In summing over the final rotational states of the molecule the change in energy of the neutron is assumed negligibly small compared to the neutron's initial energy. The validity of this approximation is discussed. The rigidity of the molecule is then relaxed and the expression obtained for the cross section amended to take account of the vibrational motion of the nuclei. It is shown that the elastic scattering from a vibrating molecule is equivalent to that from a set of uncoupled anisotropic oscillators placed at the mean nuclear sites. Detailed results are given for spherical top molecules of the type YX4. Good agreement is found with the experimental pattern of carbon tetrafluoride but not with that of methane.