Many recent language models (LMs) of the Transformers family are capable of in-context learning (ICL), manifested in the LMs' ability to perform a new task solely from its description in a natural language input. Previous work curating these models assumes that ICL emerges from vast over-parametrization or the scale of multi-task training, but recent theoretical work attributes ICL emergence to training data properties, creating in-context learners with small, synthetic data.Inspired by these findings, we propose Concept-aware Training (CoAT), a framework for constructing training scenarios that make it beneficial for the LM to learn to utilize the analogical reasoning concepts from demonstrations. We find that by using CoAT, pre-trained transformers can learn to better utilise new latent concepts from demonstrations and that such ability makes ICL more robust tofunctional deficiencies of the previous models. Finally, we show that concept-aware in-context learning improves ICL performance on a majority of new tasks compared to traditional instruction tuning, reaching performance comparable to the multitask learners using magnitudes of more training data.