Musical improvisation is a sophisticated cognitive process that combines creativity, goal-directed action, sensory monitoring and social interaction. With a renewed interest in quantifying creative processes facilitated by recent advances in neuroimaging technology, musical improvisation has emerged as an ideal paradigm to study creativity. However, many studies isolate the top-down processes related to creativity from those related to production and auditory perception, leaving the question of how creative behaviours integrate sensory information with higher cognitive processes unanswered. The bottom-up neural correlates of music perception have been extensively quantified, comprising networks for auditory processing and parsing semantic and syntactic content. In studies of spontaneously generated music and domain-general creativity, executive control and goal-directed movement networks are added to the perceptual foundation. This review summarises previous work on music perception and improvisation and presents a conceptual model of musical improvisation with known neural correlates. We make recommendations on future directions for the study of improvisation and discuss the challenges posed by this endeavour.