Human character animation is often critical in entertainment content production, including video games, virtual reality or fiction films. To this end, deep neural networks drive most recent advances through deep learning (DL) and deep reinforcement learning (DRL). In this article, we propose a comprehensive survey on the state-of-the-art approaches based on either DL or DRL in skeleton-based human character animation. First, we introduce motion data representations, most common human motion datasets and how basic deep models can be enhanced to foster learning of spatial and temporal patterns in motion data. Second, we cover state-of-the-art approaches divided into three large families of applications in human animation pipelines: motion synthesis, character control and motion editing. Finally, we discuss the limitations of the current state-of-the-art methods based on DL and/or DRL in skeletal human character animation and possible directions of future research to alleviate current limitations and meet animators' needs.