Wright was born in Martins Ferry, Ohio two years before the American stock market crash of 1929 to a father who worked in a glass factory and a mother who worked in laundry. He graduated from high school in 1946, one year after the end of World War I. Wright then joined the army and was stationed in Japan during the American occupation of that country.
Wright later attended Kenyon College], from which he graduated cum laude, after which he received a Fulbright Fellowship and travelled to Austria. After receiving his Ph.D. from the University of Washington, Wright taught at various institutes around the country, including Macalester College and the University of Minnesota.
Wright's early poetry is relatively conventional, in form and meter, especially compared with his later, looser poetry. His poetry often deals with the disenfranchised, or the outsider, American; yet it is also often inward probing.
Wright studied under American poets Theodore Roethke and Stanley Kunitz.
His 1972 Collected Poems were awarded the Pulitzer Prize. In addition to his other awards, Wright received a grant from the Rockefeller Foundation.
Wright died shortly after being diagnosed with cancer of the tongue.