Learning to play a musical instrument is a complex task that integrates multiple sensory modalities and higher-order cognitive functions. Therefore, musical training is considered a useful framework for the research on training-induced neuroplasticity. However, the classical nature-or-nurture question remains, whether the differences observed between musicians and non-musicians are due to predispositions or result from the training itself. Here we present a review of recent publications with strong focus on experimental designs to better understand both brain reorganization and the neuronal markers of predispositions when learning to play a musical instrument. Cross-sectional studies identified structural and functional differences between the brains of musicians and non-musicians, especially in regions related to motor control and auditory processing. A few longitudinal studies showed functional changes related to training while listening to and producing music, in the motor network and its connectivity with the auditory system, in line with the outcomes of cross-sectional studies. Parallel changes within the motor system and between the motor and auditory systems were revealed for structural connectivity. In addition, potential predictors of musical learning success were found including increased brain activation in the auditory and motor systems during listening, the microstructure of the arcuate fasciculus, and the functional connectivity between the auditory and the motor systems. We show that “the musical brain” is a product of both the natural human neurodiversity and the training practice.