Fog networking has become an established architecture addressing various applications with strict latency, jitter, and bandwidth constraints. Fog Nodes (FNs) allow for flexible and effective computation offloading and content distribution. However, the transmission of computational tasks, the processing of these tasks, and finally sending the results back still incur energy costs. We survey the literature on fog computing, focusing on energy consumption. We take a holistic approach and look at energy consumed by devices located in all network tiers from the things tier through the fog tier to the cloud tier, including communication links between the tiers. Furthermore, fog network modeling is analyzed with particular emphasis on application scenarios and the energy consumed for communication and computation. We perform a detailed analysis of model parameterization, which is crucial for the results presented in the surveyed works. Finally, we survey energy-saving methods, putting them into different classification systems and considering the results presented in the surveyed works. Based on our analysis, we present a classification and comparison of the fog algorithmic models, where energy is spent on communication and computation, and where delay is incurred. We also classify the scenarios examined by the surveyed works with respect to the assumed parameters. Moreover, we systematize methods used to save energy in a fog network. These methods are compared with respect to their scenarios, objectives, constraints, and decision variables. Finally, we discuss future trends in fog networking and how related technologies and economics shall trade their increasing development with energy consumption.