A large part of the theory of the iteration of functions of several variables is illustrated by the relatively simple case of functions of one variable, and it is to the study of this case that the present paper is devoted. Several distinct classes of topics immediately suggest themselves in this connection. One of these classes comprises subjects of a purely formal nature, and is concerned only with expansions in the neighborhood of certain significant points. Another class of topics deals with the iteration of a real function within an interval. Still a third treats of the total behavior of the transforming function B (x), when the given function A1 (x) is analytic and defined for all values of x. Except for certain simple cases the function B(x) cannot be one-valued. The present paper deals almost exclusively with the first two of the three classes of subjects just mentioned. In the formal problem, certain new points of view, and new results and formulke are obtained, although a large proportion of this part is a systematization of results already known but largely unrelated. The second part of this paper deals with the iteration of a real function, a question which seems to have been neglected up to date.* The third topic mentioned above, has not here been touched upon, except in the case of certain simple rational functions. A few known theorems concerning the iteration of functions of a single variable, have been omitted, in view of the fact that they seem to have little significance in connection with the subject of the iteration of functions of several variables. Only those subjects have here been discussed which appear as natural introductory material to the more general case. For references to those parts of the formal problem which have already been treated in the literature, one may consult the following: Pincherle, on Functional Equations and Operations, in the Encyklopadie d. Math. Wiss., II, A 11, and Encyclopedie d. Sci. Math., II, 26. E. Schroeder, Math. Ann., 2 (1870), p. 317, and 3 (1871), p. 296. J. Farkas, Journ. de Math. (3), 10 (1884), p. 102. G. Koenigs, Bull. Sc. Math. (2), 7 (1883), p. 340, and (3) 1 (1884), suppl., p. 14.