This paper explores energy-efficient communication for unmanned aerial vehicles (UAVs) by optimizing their trajectories. The authors derive a theoretical model for the propulsion energy consumption of fixed-wing UAVs, which depends on the UAV's velocity and acceleration. They show that both rate-maximization and energy-minimization designs are energy-inefficient in general. Instead, they propose a practical circular trajectory that optimizes the UAV's flight radius and speed to maximize energy efficiency. Additionally, an efficient algorithm is developed to maximize energy efficiency under general trajectory constraints, including initial/final locations, velocities, and maximum speed and acceleration. Numerical results demonstrate that the proposed designs achieve significantly higher energy efficiency compared to other benchmark schemes.This paper explores energy-efficient communication for unmanned aerial vehicles (UAVs) by optimizing their trajectories. The authors derive a theoretical model for the propulsion energy consumption of fixed-wing UAVs, which depends on the UAV's velocity and acceleration. They show that both rate-maximization and energy-minimization designs are energy-inefficient in general. Instead, they propose a practical circular trajectory that optimizes the UAV's flight radius and speed to maximize energy efficiency. Additionally, an efficient algorithm is developed to maximize energy efficiency under general trajectory constraints, including initial/final locations, velocities, and maximum speed and acceleration. Numerical results demonstrate that the proposed designs achieve significantly higher energy efficiency compared to other benchmark schemes.