Additive manufacturing technology has the potential to bring about significant technological changes to polymer matrix composites. In comparison to traditional manufacturing processes, 3D printing technology is being increasingly utilized in a variety of industries, including aerospace, vehicles, transportation, and medical engineering. The advantages of this technology include low cost, rapid prototyping, and the ability to manufacture complex structures. Nevertheless, the mutual influence and constraints of process parameters in the manufacturing process have led to the fact that the properties of 3D-printed composites have not been able to fully meet the requirements of practical applications. This paper focuses on the current development of the 3D printing process for fiber-reinforced thermoplastic composites. This paper especially reviews the research on the influence of Fused Filament Fabrication (FFF) process parameters (printing temperature, printing speed, ply thickness, and printing path, among others) on the mechanical properties of 3D-printed continuous fiber-reinforced thermoplastic composites. It discusses the problems brought about by the process parameters in the manufacturing process and proposes possible solutions. Finally, it considers the future development prospects and research directions of additive manufacturing of thermoplastic composites.