How to reduce power consumption within individual data centers has attracted major research efforts in the past decade, as their energy bills have contributed significantly to the overall operating costs. In recent years, increasing research efforts have also been devoted to the design of practical power-saving techniques in content delivery networks (CDNs), as they involve thousands of globally distributed data centers with content server clusters. In this paper, we present a comprehensive survey on existing research works aiming to save power in data centers and content delivery networks that share high degree of commonalities in different aspects. We firstly highlight the necessities of saving power in these two types of networks, followed by the identification of four major power-saving strategies that have been widely exploited in the literature. Furthermore, we present a high-level overview of the literature by categorizing existing approaches with respect to their scopes and research directions. These schemes are later analyzed with respect to their strategies, advantages and limitations. In the end, we summarize several key aspects that are considered to be crucial in effective power-saving schemes. We also highlight a number of our envisaged open research directions in the relevant areas that are of significance and hence require further elaborations.