Concepts play a pivotal role in various human cognitive functions, including learning, reasoning and communication. However, there is very little work on endowing machines with the ability to form and reason with concepts. In particular, state-of-the-art large language models (LLMs) work at the level of tokens, not concepts. In this work, we analyze how well contemporary LLMs capture human concepts and their structure. We then discuss ways to develop concept-aware LLMs, taking place at different stages of the pipeline. We sketch a method for pretraining LLMs using concepts, and also explore the simpler approach that uses the output of existing LLMs. Despite its simplicity, our proof-of-concept is shown to better match human intuition, as well as improve the robustness of predictions. These preliminary results underscore the promise of concept-aware LLMs.