States using runoffs for statewide or federal office
The following states use runoffs for statewide or federal offices.

 50% threshold in primaries
Arkansas  50% threshold in primaries
Georgia  50% threshold in primaries and 50% threshold for general elections for certain statewide offices
Louisiana 50% threshold in federal primaries and conditional runoff with 50% threshold in "cajun primary" system for state elections like governor
Mississippi  50% threshold in primaries. Gubernatorial candidates must surpass 50% in general elections or the legislature selects the winners.
North Carolina 40% threshold in primaries
Oklahoma  50% threshold in primaries
South Carolina  50% threshold in primaries
South Dakota 35% threshold for primaries for federal office and governor
Texas  50% threshold in primaries
 Vermont  No federal runoffs, but candidates for governor and certain other statewide offices must surpass 50% to avoid legislature electing winner.
State and federal system where two candidates advance from primary to general election.

Recently abandoned  
Florida Before 2006 elections eliminated primary runoff law after suspending it before  2002 and 2004 elections
Kentucky In 2008 repealed runoff with 40% threshold that was for governor only, not for federal office