Joseph Henry Bunce (December 15, 1823 - February 27, 1908) was the nineteenth Mayor of Louisville, Kentucky, from 1869 to 1870.
Typical of Louisville steamboat captains, he lived with his family in the Portland district.
A Democrat, he ran against future mayor John G. Baxter and police chief Robert Gilchriest in 1869.
Despite not being endorsed by the Democratic Party or The Courier-Journal, Bunce won by 300 votes over Baxter.
Bunce married Josephine Mullins (1828–1882) and had four children who survived, Mary Alice, Charles William, Harriet Archer, and Marshall Halbert.