Concept-based XAI (C-XAI) approaches to explaining neural vision models are a promising field of research, since explanations that refer to concepts (i.e., semantically meaningful parts in an image) are intuitive to understand and go beyond saliency-based techniques that only reveal relevant regions. Given the remarkable progress in this field in recent years, it is time for the community to take a critical look at the advances and trends. Consequently, this paper reviews C-XAI methods to identify interesting and underexplored open ends and proposes future research directions. To this end, we consider three main directions: the choice of concepts to explain, the choice of concept representation, and how we can control concepts. For the latter, we propose techniques and draw inspiration from the field of knowledge representation and learning, showing how this could enrich future C-XAI research.