Mobile networks are becoming energy hungry, and this trend is expected to continue due to a surge in communication and computation demand. Multi-access Edge Computing (MEC), a key component of 5𝐺 and 6𝐺 networks, will entail energy-consuming services and applications, with non-negligible impact in terms of ecological sustainability. In this paper, we provide a comprehensive review of existing approaches to make edge computing networks greener, including but not limited to the exploitation of renewable energy resources, and context-awareness (e.g., user mobility, or dynamics in the traffic), analyzing their pros and cons. We hence provide an updated account of recent developments on MEC from an energetic sustainability perspective, addressing the initial deployment of computing resources, their dynamic (re)allocation (resource scheduling), as well as distributed and federated learning designs. In doing so, we highlight the energy aspects of these algorithms, advocating the need for energy-sustainable edge computing systems that are aligned with Sustainable Development Goals (SDGs) and the Paris agreement. To the best of our knowledge, this is the first work providing a systematic literature review on the efficient deployment and management of energy harvesting MEC, with special focus on the deployment, provisioning, and scheduling of computing tasks, including federated learning for distributed edge intelligence, toward making edge networks greener and more sustainable. At the end of the paper, open research avenues and challenges are identified for all the surveyed topics. CCS Concepts: • Networks → Network performance evaluation; Wireless access points, base stations and infrastructure; • Computer systems organization → Embedded systems; • Computing methodologies → Planning and scheduling.