Non-geostationary (NGSO) satellites are envisioned to support various new communication applications from countless industries. NGSO systems are known for a number of key features such as lower propagation delay, smaller size, and lower signal losses in comparison to the conventional geostationary (GSO) satellites, which will enable latency-critical applications to be provided through satellites. NGSO promises a dramatic boost in communication speed and energy efficiency, and thus, tackling the main inhibiting factors of commercializing GSO satellites for broader utilizations. However, there are still many NGSO deployment challenges to be addressed to ensure seamless integration not only with GSO systems but also with terrestrial networks. These unprecedented challenges are discussed in this paper, including coexistence with GSO systems in terms of spectrum access and regulatory issues, satellite constellation and architecture designs, resource management problems, and user equipment requirements. Beyond this, the promised improvements of NGSO systems have motivated this survey to provide the state-of-the-art NGSO research focusing on the communication prospects, including physical layer and radio access technologies along with the networking aspects and the overall system features and architectures. We also outline a set of innovative research directions and new opportunities for future NGSO research.