There are two main trends in the development of unmanned aerial vehicle (UAV) technologies: miniaturization and intellectualization, in which realizing object tracking capabilities for a nano-scale UAV is one of the most challenging problems. In this paper, we present a visual object tracking and servoing control system utilizing a tailor-made 38 g nano-scale quadrotor. A lightweight visual module is integrated to enable object tracking capabilities, and a micro positioning deck is mounted to provide accurate pose estimation. In order to be robust against object appearance variations, a novel object tracking algorithm, denoted by RMCTer, is proposed, which integrates a powerful short-term tracking module and an efficient long-term processing module. In particular, the long-term processing module can provide additional object information and modify the short-term tracking model in a timely manner. Furthermore, a position-based visual servoing control method is proposed for the quadrotor, where an adaptive tracking controller is designed by leveraging backstepping and adaptive techniques. Stable and accurate object tracking is achieved even under disturbances. Experimental results are presented to demonstrate the high accuracy and stability of the whole tracking system.