In this paper we introduce MCA-NMF, a computational model of the acquisition of multi-modal concepts by an agent grounded in its environment. More precisely our model finds patterns in multimodal sensor input that characterize associations across modalities (speech utterances, images and motion). We propose this computational model as an answer to the question of how some class of concepts can be learnt. In addition, the model provides a way of defining such a class of plausibly learnable concepts. We detail why the multimodal nature of perception is essential to reduce the ambiguity of learnt concepts as well as to communicate about them through speech. We then present a set of experiments that demonstrate the learning of such concepts from real non-symbolic data consisting of speech sounds, images, and motions. Finally we consider structure in perceptual signals and demonstrate that a detailed knowledge of this structure, named compositional understanding can emerge from, instead of being a prerequisite of, global understanding. An open-source implementation of the MCA-NMF learner as well as scripts and associated experimental data to reproduce the experiments are publicly available.