An elementary explanation is given of the group theory method using characters to determine the selection rules for the fundamental vibrations of a linear polyatomic molecule. Then the method is extended to indicate how the selection rules for the overtones and summation and difference tones are obtained. Finally, the procedure for determining the number of fundamental frequencies belonging to a particular vibration type is given. The groupsC∞vandD∞hare used as examples in the treatment since they are the only ones involved in the consideration of linear molecules.