This paper presents a real-time three-dimensional path planning algorithm for improving autonomous navigation of Unmanned Aerial Vehicles (UAVs) operating in completely unknown cluttered environments. The algorithm generates smooth paths consisting of continuous piecewise Bezier curves in real time. Specifically, a RRT-based waypoint generation algorithm is firstly proposed for the exploration of collision-free waypoints successively during flight. Besides, a novel real-time path smoothing technique is developed to generate continuous collision-free paths that satisfy the motion constraints of UAVs. This is achieved by fitting Bezier curves between consecutive waypoints based on the particle swarm optimization (PSO) algorithm. Lastly, a path selection strategy is also introduced to seek for an optimum path when multiple solutions are available. The simulation results demonstrate the superiority of the proposed real-time three-dimensional smooth path planning algorithm.