Public safety networks (PSNs) are very crucial for public protection and disaster relief (PPDR). Land mobile radio (LMR) technologies have been used extensively in the deployment of PSNs so far. LMR networks support sophisticated voice applications that can, to some extent, deal with the mission-critical nature of PPDR services. However, LMR networks lack technological advancements to support broadband (BB) applications. Due to this limitation, the attention is drawn to the long-term evolution (LTE) technology for public safety (PS) deployment as it has the potential to support various narrowband and BB applications and services. LTE-based PSNs should have strict requirements in terms of scalability, robustness, and resilience. In this survey, we will highlight the history of PSNs, including LMR and LTE-based PSNs, discuss the requirements that have to be inherited in PSNs, and examine the spectrum allocated for PS use. Moreover, we will study the architecture of LTE-based PSNs and provide deployment and migration solutions. Furthermore, voice delivery over LTE and LTE standardized solutions tailored to support PS services are discussed. Finally, rapid emergency deployment, spectrum management, priority management, and radio resource management schemes in LTE-based PSNs are discussed as well. At the end of the survey, we present PSNs simulation environment using network simulator and provide the results of multiple disaster scenarios.