A native St. Louisan, Mayhor Conway attended Saint Louis University and earned two degrees: a B.S. in Geophysical Engineering, and an M.B.A. in Business Administration. He began his career as a sales engineer and production manager for the Nooter Corporation. Elected to the Missouri General Assembly as a representative in 1966, he served in the House until 1974, when he was elected to the State Senate, in which he served until 1977. When elected mayor he was president of the A.C.I. Plastics Company.

Mayor Conway defeated former Mayor Cervantes for the Democratic nomination in 1977. In the April 5 general election he was a winner by a large margin over Republican James A. Stemmler and William Clay, a write-in candidate. Upon his election, he put up a sign saying "Under New Management" and brought in all new staff, except for Human Resources Director Robert W. Duffe.

Mayor Conway secured a $15,000,000 in federal funding for a $150 million May Department Stores shopping mall. He also pushed legislation enabling construction of new downtown office buildings and hired Colonel Leonard Griggs, Air Force retired, to head the Airport. The mayor engaged in several public disputes with the Comptroller and the board of aldermen, some of which resulted in lawsuits and threats of lawsuits by the Mayor. He did gain control over some $35,000,000 in federal Community Development Block Grants, after a bout with the Board of Aldermen. He also succeeded in getting the $25,000 City salary cap lifted.

The phasing out of Homer G. Phillips Hospital, together with his stance supporting the long considered North-South Distributor Highway, put him in conflict with North Side residents and politicians. These conflicts were largely responsible for delaying the annual budget ordinance for 1979/80 for more than eight months. It also contributed to his inability to get re-elected.

Mayor Conway is married to the former Joan C. Newman. They have five children.