Unmanned aerial vehicles (UAVs) have attracted increasing attention in wireless communications due to the high mobility. This paper investigates a fixed-wing UAV-to-UAV (U2U) communications system, with the aim of minimizing the information transmission time via proactively designing the UAV paths. First, we propose a general optimization framework for U2U communications, which covers the communication throughput requirement, interference from terrestrial transmitters, UAV maximum/minimum speeds and accelerations, and minimum U2U distance. To tackle the formulated optimization, the communication throughput constraint that contains uncertain locations of terrestrial transmitters is transformed into a deterministic expression with the aid of S-procedure, and the nonlinear equality constraints on the UAV paths are replaced by linear equality constraints with additional positive semidefinite matrix constraints. Then, we develop a path planning algorithm based on the exact penalty method and successive convex approximation. Furthermore, we design a heuristic path planning algorithm that solves the completion time minimization problem by iteratively addressing a series of throughput maximization problems. The proposed heuristic algorithm strikes a good tradeoff between the computational complexity and the achievable performance. Finally, the simulation results are presented to verify the proposed path planning algorithms under various parameter configurations.