The occurrence of natural disasters has become more frequent in recent times, making post-disaster relief management a global concern. The resultant inaccessibility to large areas and damage to communication infrastructure hinders the optimized prioritization of relief measures. While unmanned aerial vehicles (UAVs) have played a limited role in disaster management operations, leveraging aerial edge computing (AEC) in disaster relief management supply chains has the potential to significantly enhance the efficiency and effectiveness of relief efforts. AEC combines the versality of aerial platforms with real-time data processing and analysis, thus enabling quick data driven actions. While academic interest in this field of AEC is noteworthy with an increase in the number of academic publications annually, such new innovations tend to remain in the halls of academic, at a distance from the on-the-ground decision makers and practitioners. Thus, the aim of this study was to conduct a systematic literature review of the topic to bring to the fore current understandings, challenges, and possibilities for AEC adoption. The PRISMA methodology was used to execute the systematic literature review and thematic analysis was used to extract the dominant themes. The themes that were extracted include: rapid needs assessment, logistics optimization, dynamic resource allocation, network establishment, personnel and asset tracking. The themes around challenges revolve around platform stability, safety and security, regulatory compliance, and interoperability. Despite the challenges positive sentiments resonate with  the view that together with regulations, best practices are likely to overcome challenges.