The electrochemical behaviour of microdisc electrodes coated with electroactive polymer thin films is examined theoretically. Attention is focused on the development of a theoretical model which seeks to describe radial diffusion and bimolecular chemical reaction within polymer-modified ultramicroelectrodes which specifically considers the possibility that the polymer film may overcoat the inlaid microdisc support surface. The boundary value problem is formulated, and approximate expressions for the steady-state amperometric response are presented. A particularly important parameter introduced is ζ which defines the ratio of the radius of the inlaid microdisc to the thickness of the polymer layer. The overall kinetic behaviour is mapped in terms of a kinetic case diagram. It is shown that when the radius of the inlaid microdisc is much larger than the thickness of the polymer film (this corresponds to ζ = a/L being very large), planar diffusion of substrate within the layer may be substituted in place of radial diffusion, and the theoretical analysis reduces to that previously reported by Andrieux et al., which considers simple planar diffusion and chemical reaction within a polymer-modified electrode. One refers to this observation as the large ζ limit correspondence principle. Indications as to the manner in which the ideas presented in the theoretical analysis may be examined experimentally are presented. Characteristic diagnostic plots involving the relationship between the catalytic current density and the electrode radius are developed. It is indicated that, in the quantitative investigation of heterogeneous redox catalysis at polymer-modified electrodes, examination of the manner in which the inverse current density varies with changes in the electrode radius, is equivalent to the more well established double reciprocal Koutecky-Levich analysis of inverse catalytic current density with inverse square root rotation speed that is used with polymer-coated rotating disc electrodes. Hence electrode radius replaces electrode rotation speed as an experimental variable. This makes ultramicroelectrodes very attractive candidates for experimental studies on heterogeneous redox catalysis. The situation where the substrate reacts directly at the underlying inlaid disc electrode is also considered.