He received a Bachelor of Science degree from the United States Naval Academy in 1956 and was a United States Air Force Lieutenant from 1956 to 1960, remaining in the United States Air Force Reserve from 1960 to 1970.
He received a Bachelor of Laws from Washburn University School of Law in 1963, and entered private practice in Santa Fe, New Mexico until 1964, and then in Alamogordo, New Mexico until 1980.
He was a New Mexico state senator from 1970 to 1980, serving as minority leader of the New Mexico State Senate from 1972 to 1980.
[1] On May 14, 1986, Conway was nominated by President Ronald Reagan to a seat on the United States District Court for the District of New Mexico vacated by Judge Bobby Baldock.
He served as Chief Judge from 1994 to 2000, assuming senior status on September 1, 2000.