In the realm of edge computing, the optimization of latency, energy, bandwidth, and local computation is critical, especially for mission-critical applications in sectors like disaster management and healthcare. Such applications, exemplified by deploying UAVs and autonomous robots, demand instantaneous data processing. Given the inherent constraints of edge servers—characterized by their limited capacity—meticulous resource management becomes paramount. This entails judicious resource allocation, astute provisioning, strategic task offloading, and judicious application placement, all pivotal for both fixed and mobile resource service delivery. This survey delves deep into the nuances of deploying mission-critical applications in an edge environment, dissecting their technological prerequisites. Our exploration employs a systematic literature review grounded in a conventional review methodology. We analyze the cornerstone quality of service metrics pivotal for such critical applications in edge contexts, aiming for efficient service delivery. Moreover, we identified some major gaps in current resource management strategies. Our overarching ambition is to pave the way for robust edge computing paradigms tailored for mission-critical applications.