This paper examines natural convection in the shallow annular gap between two concentric circular cylinders. Asymptotic solutions are obtained in the limit as the aspect ratio epsilon (defined as the ratio of the enclosure height to the gap width) goes to 0. It is shown that the solution at O(epsilon(n)) can only be completely specified by examining the governing equations at O(epsilon(n+2)). Solutions are obtained, and Nusselt number correlations are presented, when the dimensionless radius of the inner cylinder delta is of O(1/epsilon) and when delta is of O(1). The results indicate that curvature effects profoundly influence the nature of convection in shallow annular enclosures.