In this paper we shall consider certain simple topological properties of periodic homeomorphisms of a space S into itself. Two homeomorphisms T and To of S into itself are said to belong to the same topological type if there is a third homeomorphism r such that To = rTT 1. For arbitrary T's, many characteristic properties of types can be described in terms of the structure of certain invariant sets, the recurrence properties under repeated iteration and so on, (see for example [1]); but the problem of complete classification of types even for the case where S is, say, a sphere is too general to be of great interest. If, however, one considers only homeomorphisms of finite period, various algebraic type invariants can be defined. For orientable surfaces, for example, such invariants are actually sufficient to enable one to determine whether or not two sense-preserving periodic transformations belong to the same type (Nielsen [6]). We shall here consider such invariants as can be readily defined in terms of homology theory. For the most part, we shall maintain a purely combinatorial point of view throughout and shall take S to be a finite simplicial complex and T to be a homeomorphism of S into itself which carries simplexes into simplexes, T being then necessarily of finite period. In the definition of type, r is also understood to be a simplicial homeomorphism. It will be seen that some of our considerations resemble those which have recently led Reidemeister and de Rham to the (strictly combinatorial) classification of lensand cyclic-spaces. While our invariants do not have this sharpness, they do have the merit of being more topological in character; some of our theorems have meaning and hold true equally well for very general spaces. This implies in particular that when S is a complex, T and r do not really need to be simplicial. Although the modifications necessary to place our results in a really topological setting will not be carried out here, it will readily be seen from our papers [7] and [8] what the required procedure would be. Our invariants are closely related, as we shall show, to the homology groups of the so-called modular space of S the space obtained by identifying points which are images of each other under powers of T. In the particular case in