Emotions are biological, chemical, and neural reactions to stimuli, influenced by crucial brain mechanisms for human survival and well-being. Music is linked to the induction of emotions with neurobiological foundations associated with the activation of structures in the limbic and paralimbic systems. The induction of emotions is also related to specific features of musical stimuli. This article aims to discuss the discrete and dimensional approaches to emotions and cognitive models of musical emotions through a literature review. The discrete approach argues that certain emotional categories are universal and innate, reflecting internally triggered states driven by genetically programmed bodily changes. The dimensional approach involves models that represent emotions in a few dimensions rather than specific emotional states. Both guide models of musical emotions and enable testing hypotheses and creating emotional assessment tools with music.