Unmanned Aerial Vehicles (UAVs), commonly known as drones, show great potential in advancing the United Nations (UN) Sustainable Development Goals (SDGs), a collection of 17 worldwide targets outlined by the United Nations (UN) to tackle diverse social, economic, and environmental issues and foster sustainable development by 2030. As per the Federal Aviation Administration (FAA), UAVs can significantly improve areas of sustainability like wildlife and wildfire monitoring, precision agriculture, as well as healthcare. Furthermore, drones with communication capabilities are seen as key elements of non-terrestrial networks, and as enabling technologies for future 6G communication networks. This systematic literature review focuses on the role of UAV-enabled communications to achieve the SDGs. Our study is based on a thorough selection and analysis of scientific articles. Initially, we performed a broad search across three major databases: IEEE Xplore, Web of Science, and Scopus, yielding a set of 10,499 papers. Following a strict selection methodology (Preferred Reporting Items for Systematic Review and Meta-Analysis - PRISMA), we narrowed this set down to 28 key journal articles. Our analysis highlights six global targets that greatly benefit from UAV-enabled aerial communication. In order of impact, these are: “Sustainable Cities and Communities,” where UAVs improve security and disaster management; “Life on Land,” where drones help with wildlife monitoring and reforestation; “Industry, Innovation, and Infrastructure,” where UAVs (Unmanned Aerial Vehicles) aid in infrastructure inspection; “No Hunger,” where drones improve agricultural practices; “‘Good Health and Wellbeing,” where UAVs assist with disease prediction and monitoring; and “Clean water and sanitation,”where drones play a crucial role in efficiently monitoring water quality and distribution. In summary, this paper offers a detailed look at the applications of UAVs in achieving the SDGs through aerial communications. By identifying areas of major impact and spotting gaps and weaknesses in current research, this review article aims to guide future research and policy-making in sustainable drone applications.