Producing and perceiving music engage a wide range of sensorimotor, cognitive, and emotional processes. Emotions are a central feature of the enjoyment of music, with a large variety of affective states consistently reported by people while listening to music. However, besides joy or sadness, music often elicits feelings of wonder, nostalgia, or tenderness, which do not correspond to emotion categories typically studied in neuroscience and whose neural substrates remain largely unknown. Here we review the similarities and differences in the neural substrates underlying these "complex" music-evoked emotions relative to other more "basic" emotional experiences. We suggest that these emotions emerge through a combination of activation in emotional and motivational brain systems (e.g., including reward pathways) that confer its valence to music, with activation in several other areas outside emotional systems, including motor, attention, or memory-related regions. We then discuss the neural substrates underlying the entrainment of cognitive and motor processes by music and their relation to affective experience. These effects have important implications for the potential therapeutic use of music in neurological or psychiatric diseases, particularly those associated with motor, attention, or affective disturbances.