When we listen to music, we often feel the urge to tap our fingers, stomp our feet, or move our bodies. This pleasurable sensation of wanting to move one's body with the music has been termed "groove." Research on groove was initially conducted mainly in the fields of musicology, ethnomusicology and philosophy, and in the 2000s, researchers within the field of empirical psychology began to investigate the concept of groove and the characteristics of music associated with it. In the 2010s, studies of the relationship between groove and underlying brain structures and functional processes emerged and the topic has recently become a prominent theme in empirical research on musical behavior. This review aims to provide a comprehensive characterization of groove and to highlight future directions for related empirical research. In doing so, we focus on two topics that have the potential to provide new insights into perception and action from the perspectives of psychology and neuroscience: (1) sensorimotor interactions and (2) the role of the motor system in predictive processing. To this end, we survey the literature on groove conducted in the fields of psychology and neuroscience from 2006 to 2022 in five sections: (1) the concept of groove, (2) musical features related to groove, (3) responses to groove including body movement and neurophysiological activity, (4) characteristics of listeners that influence the experience of groove, and (5) the effect of culture and environment on groove perception. Through the review, we propose four themes as promising future directions for empirical study: (1) social aspects of groove, (2) the effect of microtiming and syncopation on groove, (3) the mechanisms by which groove is evoked, and (4) clinical implications and applications of groove. Taken together, these research efforts may ultimately lead to an increased understanding of human sociality, evolution, and the brain and mind.