The 5G communication systems will enable enhanced mobile broadband, ultra-reliable low-latency, and massive connectivity services. Broadband and low-latency services are indispensable to public safety (PS) communication during natural or man-made disasters. Recently, 3GPP-LTE has emerged as a promising candidate to enable broadband PS communications. In this article, first we present six major PS-LTE enabling services and the current status of PS-LTE in 3GPP releases. Then, we discuss the spectrum bands allocated for PS-LTE in major countries by ITU. Finally, we propose a disaster resilient three-layered architecture for PS-LTE (DR-PSLTE). This architecture consists of an SDN layer to provide centralized control, UAV cloudlet layer to facilitate edge computing or to enable the emergency communication link, and a radio access layer. The proposed architecture is flexible and combines the benefits of SDNs and edge computing to efficiently meet the delay requirements of various PS-LTE services. Numerical results verified that under the proposed DR-PSLTE architecture, delay is reduced by 20 percent as compared with the conventional centralized computing architecture.