Wright was born near Ludlow, in Shropshire, descended from a Quaker family formerly living at Bradford, Yorkshire. He was educated at Ludlow Grammar School and at Trinity College, Cambridge, where he graduated in 1834.

While at Cambridge he contributed to the Gentleman's Magazine and other periodicals, and in 1835 he came to London to devote himself to a literary career.

His first separate work was Early English Poetry in Black Letter, with Prefaces and Notes (1836, 4 vols. 12mo), which was followed during the next forty years by an extensive series of publications, many of lasting value. He helped to found the British Archaeological Association and the Percy Society, the Camden and the Shakespeare Society. In 1842 he was elected corresponding member of the Académie des Inscriptions et Belles Lettres of Paris, and was a fellow of the Society of Antiquaries as well as member of many other learned British and foreign bodies.

In 1859 he superintended the excavations of the Roman town of Viroconium Cornoviorum (Wroxeter), near Shrewsbury, and issued a report.

A portrait of him is in the Drawing Room Portrait Gallery for October 1, 1859.

He was a great scholar, but will be chiefly remembered as an industrious antiquary and the editor of many relics of the Middle Ages.

He died at Chelsea, London in his 67th year. He is buried in Brompton Cemetery.