Curtis was born in Carmarthen and educated at the University of Wales, Swansea. He was given a Gregory Award in 1972. He was winner of the National Poetry Competition in 1984 and the Dylan Thomas Award in 1993. In 1994 he became Professor of Poetry at the University of Glamorgan. In 1997 he received a Cholmondeley Award. He founded Creative Writing as a subject at the university and developed the Masters course there which he ran for sixteen years:the M.Phil. in Writing. He was awarded the first D.Litt. by the University of Glamorgan in 2004. In 2001 he was made a Fellow of the Royal Society of Literature.