As renewable energy becomes increasingly popular in the global electric energy grid, improving the accuracy of renewable energy forecasting is critical to power system planning, management, and operations. However, this is a challenging task due to the intermittent and chaotic nature of renewable energy data. To date, various methods have been developed, including physical models, statistical methods, artificial intelligence techniques, and their hybrids to improve the forecasting accuracy of renewable energy. Among them, deep learning, as a promising type of machine learning capable for discovering the inherent nonlinear features and high-level invariant structures in data, has been frequently reported in the literature. This paper provides a comprehensive and extensive review of renewable energy forecasting methods based on deep learning to explore its effectiveness, efficiency and application potential. We divide the existing deterministic and probabilistic forecasting methods based on deep learning into four groups, namely deep belief network, stack auto-encoder, deep recurrent neural network and others. We also dissect the feasible data preprocessing techniques and error post-correction methods to improve the forecasting accuracy. Extensive analysis and discussion of various deep learning based forecasting methods are given. Finally, we explore the current research activities, challenges and potential future research directions in this topic.