This paper deals with the time-varying formation tracking control problem for multiple unmanned aerial vehicle (multi-UAV) systems, where the UAVs are required to form a predefined time-varying formation while tracking the trajectory of the leader. A finite-time formation tracking control law is designed to realize time-varying formation tracking by utilizing nonsingular fast terminal sliding mode (NFTSM), which can speed up the convergence rate and be free from singularity problem. The finite-time stability and robustness of the close-loop system is proven using Lyapunov stability theory. Finally, numerical simulation results verify the effectiveness of the proposed control law.