In mission critical situations, communications among responders must maintain a high level of reliability, availability, quality of service and security in scenarios that go beyond the limits that are stipulated in a commercial wireless network. The 3rd Generation Partnership Project (3GPP) Technical Specification Group on Service and System Aspects (TSG-SA) has developed a series of technical specifications in its SA6 working group to define an architecture for various mission critical services including mission critical push to talk (MCPTT), mission critical data (MCData), and mission critical video (MCVideo). This paper provides an overview of 3GPP standardization of mission critical services in LTE and its evolution in the 5th generation (5G) wireless networks.