He was educated at Beccles Grammar School and Trinity College, Cambridge, where he graduated Bachelor in 1858.
As a nonconformist, Wright was ineligible for election to a Trinity fellowship until 1878, but became Librarian and Senior Bursar of Trinity before that date. He opposed the allegations by Simonides that the Codex Sinaiticus discovered by Constantin von Tischendorf was produced around 1840. Duly elected Fellow in 1878, he became vice-master of the college in 1888.
He was one of the editors of the Journal of Philology from its foundation in 1868, and was secretary to the Old Testament revision company from 1870 to 1885.
He edited the plays of Shakespeare published in the "Clarendon Press" series (1868-1897), also with West. G. Clark the "Cambridge" Shakespeare (1863–1866. 2nd ed 1891–1893) and the "Globe" edition (1864).
He published a facsimile of the Milton manuscript in the Trinity College library (1899), and edited Milton"s poems with critical notes (1903). This was followed by the Letters of Edward Fitzgerald to Fanny Kemble (1895), his Miscellanies (1900), More Letters of Edward Fitzgerald (1901), and The Works of Edward Fitzgerald (7 vols, 1903).
He edited the metrical chronicle of Robert of Gloucester (1887), Generydes (1878) for the Early English Text Society, Catalogue of the Syriac manuscripts in the British Museum (1–3 volume, 1870–1872), and other texts.
He is buried in the Parish of the Ascension Burial Ground in Cambridge.