English composer, teacher, theorist, and one of the great virtuoso players in the history of the viol.

A Roman Catholic, he fought on the Royalist side in the English Civil War (1643–44) and subsequently became tutor to the son of a prominent Catholic, Sir Robert Bolles. During his life Simpson was highly regarded as a musician. As a composer, chiefly of solo and ensemble works for viol, he is noteworthy for his exploitation of the instrument's resources and his development of variation form. His influential theoretical works were The Division-violist (1st ed., 1659; modern ed., 1955, reprinted 1998), discussing viol technique and the improvisation of descants and divisions (variations on a ground); and The Principles of Practical Musick (1665; modern ed., 1970), praised for its excellence by Henry Purcell and other contemporary composers.