The number of disasters has increased over the past decade where these calamities significantly affect the functionality of communication networks. In the context of 6G, airborne and spaceborne networks offer hope in disaster recovery to serve the underserved and to be resilient in calamities. Therefore, this paper surveys the state-of-the-art literature on post-disaster wireless communication networks and provides insights for the future establishment of such networks. In particular, we first give an overview of the works investigating the general procedures and strategies for counteracting any large-scale disasters. Then, we present the possible technological solutions for post-disaster communications, such as the recovery of the terrestrial infrastructure, installing aerial networks, and using spaceborne networks. Afterward, we shed light on the technological aspects of post-disaster networks, primarily the physical and networking issues. We present the literature on channel modeling, coverage and capacity, radio resource management, localization, and energy efficiency in the physical layer and discuss the integrated space-air-ground architectures, routing, delay-tolerant/software-defined networks, and edge computing in the networking layer. This paper also presents interesting simulation results which can provide practical guidelines about the deployment of ad hoc network architectures in emergency scenarios. Finally, we present several promising research directions, namely backhauling, placement optimization of aerial base stations, and the mobility-related aspects that come into play when deploying aerial networks, such as planning their trajectories and the consequent handovers.