Voters are selecting major party candidates for both of the state's U.S. Senate seats in addition to deciding crowded races for lieutenant governor and superintendent of education. The polls are open from 7 a.m. to 7 p.m. EDT.
Graham is seeking the GOP nomination for a third term but must get by six GOP challengers who say he's not conservative enough.
Graham, a prohibitive favorite, has been able to draw on a $7 million campaign fund. The question is whether he will be able to win the nomination in a seven-way race without a runoff. He needs more than half the vote to do so.
In an unusual election year, voters are also selecting nominees for the state's other U.S. Senate seat — the one held by Republican Tim Scott, appointed in 2012 to replace Jim DeMint, who resigned. Scott, in his first statewide race, is seeking the rest of DeMint's unexpired term.
There are no primaries in the biggest race of the year, the gubernatorial contest pitting incumbent Republican Nikki Haley against Democratic state Sen. Vincent Sheheen in a rematch of their 2010 race.
There's a wild card this time in the governor's race: Tom Ervin of Greenville, who says he's running as an independent Republican, delivered his nominating petitions to the State Election Commission on Monday.
Eight Republicans are seeking the nomination for state superintendent of education. They include Sally Atwater, the widow of the late, well-known Republican strategist Lee Atwater. Four Democrats are seeking their party nomination.
The four-way GOP race for lieutenant governor includes, among others, former state party chairman Henry McMaster and Mike Campbell, the son of the popular late South Carolina Gov. Carroll Campbell. The winner of Tuesday's Republican primary faces Democratic state Rep. Bakari Sellers in November.
It's the last time voters will elect a lieutenant governor. Beginning in 2018, the lieutenant governor and governor will run as a ticket.
Five congressional nominations would be decided Tuesday. Incumbent Republican U.S. Rep. Joe Wilson and Democratic U.S. Rep. Jim Clyburn are being challenged by little-known candidates.