As a key technology for the success of space missions, spacecraft attitude and orbit control is typically nonlinear, and the essence is to control the dynamic system evolving on the matrix Lie groups SO(3) and SE(3). Compared with the traditional parameterized model, the matrix Lie group model of a spacecraft has some inherent advantages on the representation of attitude and orbit, such as globality, non-singularity, and uniqueness, which provides the model basis of mathematical simplicity, high precision and good applicability. In recent years, direct research on its system analysis and control design based on differential geometric control methods is gradually emerging, and a series of breakthrough achievements have been made. Thus, this paper summarizes the research results of the application of differential geometric control theory in spacecraft attitude and orbit control. Facing the requirements of actual space missions, the three technical directions of attitude control, orbit control and networked spacecraft cluster coordinated control based on the matrix Lie group model are discussed respectively. Finally, the challenges in the research field are summarized, while the future developments along the direction are prospected.