Conway was born in Jonesboro, Arkansas and attended Southeast Missouri State University, graduating in 1969. He was commissioned as an infantry officer in 1970. His first assignment was command of a rifle platoon with 3rd Battalion, 1st Marines out of Camp Pendleton. He also served as the Battalion's 106mm recoilless rifle platoon commander. Later he served as Marine Executive Officer aboard the aircraft carrier USS Kitty Hawk.
After graduating with honors from career-level officer school, Conway commanded two companies in the 2nd Marine Regiment's Operations and Security section, later commanding two companies at Marine Basic School. He then went on to serve as operations officer for the 31st Marine Amphibious unit, with sea duty in the western Pacific and in contingency operations off Beirut, Lebanon.
Returning to the U.S., Conway was assigned as Senior Aide to the Chairman of the Joint Chiefs of Staff, serving two years in that capacity. After graduating from top-level officer training, again with honors, he took command of 3rd Battalion, 2nd Marines, and commanded the Battalion Landing Team in its eight-month deployment to Southwest Asia during Desert Storm.
After the war he was promoted to colonel, and assigned command of Marine Basic School. Promoted to Brigadier General in December 1995, he again was assigned to the JCS. After being promoted to Major General, he served as commander of the 1st Marine Division and as Deputy Commanding General of Marine Forces Central. He was promoted to Lieutenant General and assumed his current assignment with I MEF on November 16, 2002.
In a press interview on May 30, 2003, General Conway was questioned about the failure at that point to locate weapons of mass destruction in Iraq. He replied, in part: