A highly effective control method is very important to guarantee the safety of the formation of flying missions for multiple unmanned aerial vehicles (UAVs), especially in the presence of complex flying environments and actuator constraints. In this regard, this paper investigates the formation tracking control problem of multiple UAVs in the presence of actuator saturation. Firstly, a brand-novel finite-time anti-saturated control scheme is proposed for multiple UAVs to track the desired position commands, wherein the tracking performance is tuned by introducing a logarithmic function-based state-mapping policy. Then, an adaptive scheme based on projection rules is devised to compensate for the negative effects brought by the actuator saturation. Based on the proposed formation tracking controller, the finite-time formation tracking performance tuning and control saturation problems can be addressed simultaneously with a comparatively allowable system robustness. Finally, three groups of illustrative examples are organized to verify the effectiveness of the proposed formation tracking control scheme.