This paper presents a concise overview of current public safety communication networks known as LMR (Land Mobile Radio) and emerging LTE-(Long-Term Evolution-) based broadband public safety networks to be deployed in the 700 MHz band. A broadband nationwide network for public safety based on LTE is inevitable where shared or dedicated types of LTE-based public safety networks are possible. Current LTE services do not meet mission-critical requirements and several enhancements have been defined by 3GPP to address this in Releases 12 and 13. First responders are familiar with LMR and consider it to be a reliable technology with massive deployment everywhere. Therefore, it is expected that LMR will continue to exist alongside any new LTE-based broadband public safety network. Recent LTE releases (particularly Release 15) addressed the LMR-LTE interoperability issue and described comprehensive interworking facilities. New and upcoming features and services of LTE in Releases 14 and 15, such as mission-critical data, mission-critical video, and aerial user equipments, are also directly applicable to public safety. The paper endeavours to provide a quick yet meaningful review of all these issues. It also offers a look ahead at the new and rapidly advancing virtualization technologies, such as software-defined radio access network, and radio access network slicing, as enablers for future public safety networks.