Unmanned aerial systems and renewable energy are two research areas that have developed rapidly over the last few decades. Solar-powered unmanned aerial vehicles (SUAVs) are likely to become dominant in the near future. They have the advantage of low cost and safe operation features that mitigate the barriers to their use in various environments. Developing effective algorithms for navigating and deploying SUAVs is essential for implementing this technology in real-life applications. Effective navigation and deployment algorithms also ensure the safety and efficiency of SUAV operations. This comprehensive review paper summarizes some state-of-the-art SUAV applications and provides an overview of the navigation and deployment algorithms for SUAVs. Some commonly used energy-harvesting models are described as well. Finally, some interesting and promising directions for future SUAV research are suggested.