On February 16, South Carolina U.S. House member Mick Mulvaney resigned to become Director of the Office of Management & Budget. South Carolina will hold a special election in his 5th district to replace him on June 20.
The Republican Party will hold a primary to choose its nominee on May 2. If no one gets 50%, there will be a run-off primary on May 16. Candidates who want a party nomination (whether the party nominates by primary or convention) file by noon on March 13. So far four Republicans, but no Democrats, have filed. Independent candidates need 10,000 signatures by April 21.
The only minor party candidate who has filed so far is Josh Putnam of the American Party. See this story about him. Thanks to Jack Dean for the link.