Corbett has served as a teacher in the Expository Writing program at Harvard University and as Writer-in-Residence in the Program of Writing and Humanistic Studies at Massachusetts Institute of Technology (MIT), where he has taught classes focusing on the craft of the personal essay and the creation of poetry. He edits for the small publisher Pressed Wafer, which specializes in poetry broadsides, chapbooks and books.

Corbett has edited the letters of James Schuyler and published two memoirs: Philip Guston’s Late Work: A Memoir (1994) and Furthering My Education (1997). He writes frequently on art, and in 1999 published a book on the sculptor John Raimondi. Corbett has been a member of the CUE Art Foundation's Advisory Council since the Foundation opened its doors in 2003. He lives in Boston’s South End and has served as the poetry editor of Grand Street. Corbett has also been a frequent contributor to local Boston newspapers, including the Boston Phoenix, for which he writes book reviews.

He is married and has two daughters.