Effective emergency and natural disaster management depend on the efficient mission-critical voice and data communication between first responders and victims. Land mobile radio system (LMRS) is a legacy narrowband technology used for critical voice communications with limited use for data applications. Recently, long term evolution (LTE) emerged as a broadband communication technology that has a potential to transform the capabilities of public safety technologies by providing broadband, ubiquitous, and mission-critical voice and data support. For example, in the United States, FirstNet is building a nationwide coast-to-coast public safety network based on LTE broadband technology. This paper presents a comparative survey of legacy and the LTE-based public safety networks, and discusses the LMRS-LTE convergence as well as mission-critical push-to-talk over LTE. A simulation study of LMRS and LTE band class 14 technologies is provided using the NS-3 open source tool. An experimental study of APCO-25 and LTE band class 14 is also conducted using software-defined radio to enhance the understanding of the public safety systems. Finally, emerging technologies that may have strong potential for use in public safety networks are reviewed.