This paper studies energy-efficient UAV communication through trajectory optimization, considering the propulsion energy consumption of UAVs. A theoretical model for the propulsion energy consumption of fixed-wing UAVs is derived, based on which the energy efficiency of UAV communication is defined. For the case of unconstrained trajectory optimization, it is shown that both the rate-maximization and energy-minimization designs lead to vanishing energy efficiency and are therefore energy-inefficient. A practical circular trajectory with optimized flight radius and speed is introduced to maximize energy efficiency. Furthermore, an efficient algorithm is proposed for generally constrained trajectory optimization, based on linear state-space approximation and sequential convex optimization techniques. Numerical results show that the proposed designs achieve significantly higher energy efficiency than heuristic rate-maximization and energy-minimization designs for UAV communications.This paper studies energy-efficient UAV communication through trajectory optimization, considering the propulsion energy consumption of UAVs. A theoretical model for the propulsion energy consumption of fixed-wing UAVs is derived, based on which the energy efficiency of UAV communication is defined. For the case of unconstrained trajectory optimization, it is shown that both the rate-maximization and energy-minimization designs lead to vanishing energy efficiency and are therefore energy-inefficient. A practical circular trajectory with optimized flight radius and speed is introduced to maximize energy efficiency. Furthermore, an efficient algorithm is proposed for generally constrained trajectory optimization, based on linear state-space approximation and sequential convex optimization techniques. Numerical results show that the proposed designs achieve significantly higher energy efficiency than heuristic rate-maximization and energy-minimization designs for UAV communications.