Recent investigations by Butcher have co-ordinated several aspects of the theory of, and techniques for the derivation of, numerical methods for ordinary differential equations, and further have yielded several significant classes of new methods. It appears that the fruitfulness of this research is, in part, a result of the generality with which this author has approached his subject. Using the same sort of general approach, several applications and generalizations in this field of research are developed as the subject of this thesis.