South Koreans began the process of selecting a new president on Tuesday, in a vote that will turn on several crucial issues: relations with North Korea and the United States; economic inequality; and the enduring power of the country’s family-controlled conglomerates, known as chaebol. Here’s how these issues are playing out in the election.

Handling North Korea

Under the current conservative government, South Korea has taken a confrontational approach toward the North, engaging in military exercises with the United States off the peninsula and participating in tightening sanctions over the North’s missile and nuclear weapons programs. But the candidate leading in the polls, Moon Jae-in, has said he is open to a dialogue with North Korea’s leader, Kim Jong-un, over the nuclear issue, a sharp break with recent policy. Mr. Moon, the candidate for the Democratic Party of Korea, says sanctions alone are not enough to persuade the North to freeze and then dismantle its nuclear programs.

One of his rivals, Hong Joon-pyo, the candidate for the Liberty Korea Party, has said that a government under Mr. Moon would be too soft on North Korea. Mr. Hong says he is the true representative of conservatives, who favor close ties to the United States, and is calling for “armed peace” that supports the status quo of being tough on North Korea.

U.S. Relations

A defining issue has been the current government’s acceptance of an American antimissile system on South Korean soil to guard against missile attacks from the North. Mr. Moon’s main opponents — Mr. Hong and Ahn Cheol-soo, a centrist who represents the People’s Party — have expressed support for the deployment of the system, called the Terminal High Altitude Area Defense system, or Thaad. Mr. Moon, by contrast, has called the system’s recent deployment “very regrettable” and said in a book published recently that South Korea should learn to “say no to the Americans.”