Ten states require a candidate to win a primary with a majority of the votes. To make that happen, primary runoff elections are used: Alabama, Arkansas, Georgia, Mississippi, North Carolina, Oklahoma, South Carolina, South Dakota, Texas and Vermont. Vermont holds runoffs only in the event of a tie. South Dakota only holds runoffs for the offices of U.S. Senator, U.S. Representative and the governor.
Alabama: A runoff is required if no candidate gets a majority of the vote in the primary. The top two candidates in the primary go to the runoff. Runoff elections are held on Tuesday of the ninth week after the primary. For its uniform and overseas absentee voters only, Alabama uses instant runoff voting, also known as ranked-choice voting. Read Alabama's explanation of the state's process, or NCSL's page on alternative voting methods for more general information on instant runoff/ranked-choice voting.
Arkansas: As in most other states, a runoff between the two top candidates is required if no candidate gets a 50% majority in the primary. Runoff elections are normally held three weeks after the primary.
Georgia: If no candidate gets a majority of the votes cast, a runoff between the top two candidates is required. Runoff elections are held nine weeks after the primary.
Mississippi: A runoff is required between the top two candidates unless one candidate gets a majority. Runoff elections are normally held three weeks after the primary. In 2020, Mississippi voters passed Measure 2, which established runoffs for governor and other statewide offices, in addition to those already held for legislative offices.
North Carolina: A runoff (they call it a second primary) is not required if a candidate gets a "substantial" plurality–which is defined as 30% of the vote plus one. A runoff is not required in any case unless the second-highest vote-getter calls for a runoff. Runoff elections are normally held seven weeks after the primary.
Oklahoma: Provisions are the same as in most other states–a majority is required to preclude a runoff, otherwise the two top candidates go to the runoff. Runoff elections are held in August.
South Carolina: In South Carolina, the primaries are the responsibility of the political parties, but they operate the same way that most states do: a majority precludes a runoff. Runoff elections are normally held two weeks after the primary.
South Dakota: In South Dakota, a runoff is held only for the offices of U.S. Senator, U.S. Representative, or governor. This is called a “secondary” election. If, in a primary race involving three or more candidates, no candidate receives 35% of the vote, the two candidates receiving the highest number of votes move to a “secondary” election ten weeks after the first primary election.
Texas: It is the same in Texas as in many other states; getting a majority precludes a runoff; otherwise, the two top vote-getters go to a runoff. Runoff elections are normally held six weeks after the primary.
Vermont: Vermont conducts a runoff only in the event of a tie in the primary.
NOTE: Beginning in 1992, Kentucky held a primary runoff if no candidate for governor or lieutenant governor received 40% in the primary race for governor. This was repealed in 2008 (2008 Ky. Acts, Chap. 129).
NCSL's webpage on State Primary Types
NCSL's State Primary Types Table
NCSL's report, Primaries: More than One Way to Find a Party Nominee
NCSL's webinar, Beyond Primaries: The Legislative Role, produced in May 2017. The webinar covers the history of primary runoff elections, why states use them and what’s changed over the years. It includes a discussion of ranked-choice voting—a method some suggest can replace runoffs (hence the other name, “instant runoff voting”).