A self-balancing unicycle robot, which has a wheel for balancing and moving in the longitudinal plane (pitch angle) as well as a vertical flywheel for balancing in the lateral plane (roll angle), is studied in this paper. The non-linear dynamic equations of the unicycle robot on slope are analyzed using the Lagrangian dynamic formulation, then a linear model of the robot is derived at the equilibrium point, and linear quadratic regulators (LQR) are designed to control the robot on slopes with the angle of inclination varying from -11° to 11°. Simulation and physical experiment results validated that the unicycle robot can achieve good performance both on level plane and on slope. As far as we know, this is the first report of an autonomous unicycle robot moving and balancing on slope.