Besides conventional geostationary (GSO) satellite broadband communication services, non-geostationary (NGSO) satellites are envisioned to support various new communication use cases from countless industries. These new scenarios bring many unprecedented challenges that will be discussed in this paper alongside with several potential future research opportunities. NGSO systems are known for various advantages, including their important features of low cost, lower propagation delay, smaller size, and lower losses in comparison to GSO satellites. However, there are still many deployment challenges to be tackled to ensure seamless integration not only with GSO systems but also with terrestrial networks. In this paper, we discuss several key challenges including satellite constellation and architecture designs, coexistence with GSO systems in terms of spectrum access and regulatory issues, resource management algorithms, and NGSO networking requirements. Additionally, the latest progress in provisioning secure communication via NGSO systems is discussed. Finally, this paper identifies multiple important open issues and research directions to inspire further studies towards the next generation of satellite networks.