Due to rapid advancements on the Internet of Things (IoT), unmanned aerial vehicles (UAVs), also known as drones, are transforming numerous military and civil application areas. UAVs aim to enhance the production efficiency, ensure safety, and reduce risk, particularly protecting the human workforce in the case of harsh and dangerous environments. Due to the mission-critical, business-critical, or safety-critical nature of the UAV applications, it is pivotal that UAVs perform reliably to deliver the required service during the intended mission time. Therefore, reliability is one of the essential requirements for designing and operating UAVs. This article presents a critical review of UAV reliability literature in both theoretical and practical research, pinpointing failure causes and reliability challenges of UAV systems, classifying and reflecting on the reliability modeling, analysis, and design methods for UAV systems and key subsystems. Some open research problems and opportunities are also discussed to highlight potential new challenges for designing reliable and resilient UAVs and UAV-assisted IoT systems.