The first part of this paper deals with the problem of describing and measuring microstructure in exact terms. The Euclidean parameters: volume, area, length, and angle can be measured and expressed only in terms of the total of each in unit volume. Average properties, such as average grain diameter, are accessible only through the topological parameters, specifically number in unit volume, which can be measured only by serial sectioning. The parameters which have been used to represent the concept of grain size are analyzed and shown in most cases to represent a function of grain boundary area. In a second section of this paper the geometric problem of plastic slip through a grain boundary is analyzed. A method is proposed by which all of the components of the deformation, as well as the crystallographic directions, can be manipulated simultaneously through the use of a stereographic projection. The third section of this paper is concerned with the geometry of grain growth. The polycrystalline state is described as a grain boundary network, which must respond to the requirements of surface tension. The several topological changes in a network which can contribute to grain growth are described.