Neuroplasticity refers to the fact that our brain can partially modify both structure and function to adequately respond to novel environmental stimulations. Neuroplasticity mechanisms are not only operating during the acquisition of novel information (i.e., online) but also during the offline periods that take place after the end of the actual learning episode. Structural brain changes as a consequence of learning have been consistently demonstrated on the long term using non-invasive neuroimaging methods, but short-term changes remained more elusive. Fortunately, the swift development of advanced MR methods over the last decade now allows tracking fine-grained cerebral changes on short timescales beyond gross volumetric modifications stretching over several days or weeks. Besides a mere effect of time, post-learning sleep mechanisms have been shown to play an important role in memory consolidation and promote long-lasting changes in neural networks. Sleep was shown to contribute to structural modifications over weeks of prolonged training, but studies evidencing more rapid post-training sleep structural effects linked to memory consolidation are still scarce in human. On the other hand, animal studies convincingly show how sleep might modulate synaptic microstructure. We aim here at reviewing the literature establishing a link between different types of training/learning and the resulting structural changes, with an emphasis on the role of post-training sleep and time in tuning these modifications. Open questions are raised such as the role of post-learning sleep in macrostructural changes, the links between different MR structural measurement-related modifications and the underlying microstructural brain processes, and bidirectional influences between structural and functional brain changes.