In this paper we examine the steady, two-dimensional convective motion which occurs in a horizontal circular cylinder whose wall is non-uniformly heated. One observes several qualitatively different physical phenomena depending on the wall temperature distribution and the value of the Rayleigh number. The low-Rayleigh-number behaviour for the single convective cell heated from below is related to the classical Rayleigh stability problem. The critical Rayleigh number for the single circular cell is approximately five times the value for Rayleigh's multi-celluar configuration. The flow which exhibits a nearly parabolic velocity profile near the critical Rayleigh number, gradually changes to a rigidly-rotating-core behaviour as the Rayleigh number increases. The speed of core rotation is a function of the Prandtl number, whereas the boundary-layer thickness is primarily a function of the Rayleigh number. When the heating is from side to side, the solution shows that as the Rayleigh number increases the core motion is progressively arrested leaving a narrow circulating band of fluid adjacent to the wall. An oblique heating produces a hybrid phenomenon, a low-Rayleigh-number behaviour which is characteristic of the sideways heating case and a high-Rayleigh-number interior motion characteristic of the bottom heating case. To determine the core motion in the high-Rayleigh-number limit, Batchelor's work concerning the uniqueness of incompressible, exactly steady, closed streamline flows with small viscosity is extended to include flows with small thermal conductivity.