Spaceborne global navigation satellite system (GNSS) has significantly revolutionized the development of autonomous orbit determination techniques for low Earth orbit satellites for decades. Using a state-of-the-art combination of GNSS observations and satellite dynamics, the absolute orbit determination for a single satellite reached a precision of 1 cm. Relative orbit determination (i.e., precise baseline determination) for the dual satellites reached a precision of 1 mm. This paper reviews the recent advancements in GNSS products, observation processing, satellite gravitational and non-gravitational force modeling, and precise orbit determination methods. These key aspects have increased the precision of the orbit determination to fulfill the requirements of various scientific objectives. Finally, recommendations are made to further investigate multi-GNSS combinations, satellite high-fidelity geometric models, geometric offset calibration, and comprehensive orbit determination strategies for satellite constellations.