Other contenders for the top job in South Korea's election are Hong Joon-pyo, right, from the Liberty Korea Party, and Ahn Cheol-soo, with the People's Party.
Moon, now leading in the polls by almost 20 percent, is closely associated with the "Sunshine Policy" of engagement with North Korea - promoting economic aid and dialogue in hopes of building trust - which he supported as an aide to the last liberal president, Roh Moo-hyun.
The election is taking place amid a confluence of unprecedented events in South Korea's history: the biggest influence-peddling and corruption scandal that ousted Park, rising tensions on the Korean peninsula as a new US President tries to nip North Korea's nuclear threat in the bud, and frosty ties with China, which is retaliating against the deployment of a US missile defence system on South Korean soil.
South Koreans voted Tuesday for a new president, with victory widely predicted for a liberal candidate who has pledged to improve ties with North Korea, re-examine a contentious US missile defense shield and push sweeping economic changes.
A Gallup Korea poll published last Wednesday showed Moon with 38 percent support in a field of 13 candidates, with centrist Ahn Cheol-soo his nearest challenger with 20 percent.
However, the growing brinkmanship between the United States and North Korea has also influenced the elections, particularly following the deployment in the South of a U.S. missile defense system, the Terminal High Altitude Area Defense or "THAAD".
"I am the best-prepared presidential candidate who can overcome the economic, security and diplomatic crisis facing South Korea", Mr Moon proclaimed on TV.
The election is to choose a successor to Park Geun Hye after her impeachment for corruption and abuse of power.
Times have changed since liberals were last in charge in Seoul and many experts don't expect that Moon would push for any big reconciliation projects because North Korea has gone too far in its nuclear development over the last decade. "My party and I invested all our efforts with a sense of desperation, but we also felt a great desire by people to build a country we can be proud of again", Moon, 64, told reporters after casting his ballot.
South Korean presidential candidate Hong Jun-Pyo, left, prepares to give a speech during a campaign rally in Seoul on May 8. South Koreans voted Tuesday for a new president, with victory widely predicted for a liberal candidate who has pledged to improve ties with North Korea, re-examine a contentious US missile defense shield and push sweeping economic changes.
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They offered the college Republicans, who had invited Coulter, a different date and venue for the speech, but were turned down. UC Berkeley spokesman Dan Mogulof credited the peacefulness of Thursday's rallies partly to an increased police presence.
People wait in line to vote in advance at Seoul station, South Korea, on May 4, 2017. The US rushed to deploy a missile shield in South Korea shortly despite objections from Mr Moon, who wants the next leader to review the decision.
"I could sit down with Kim Jong Un ... when preconditions of resolving the nuclear issue are assured", Moon Jae-in of the liberal Democratic Party told the Washington Post in an interview.
When Roh became South Korean president in 2003, he took Moon with him to Seoul, putting him in a series of senior positions, including chief of staff.
With THAAD having gone live in South Korea during recent weeks, public division over it has burned beneath the election. She has been indicted on bribery, extortion and other corruption charges, which could send her to jail for life if she is convicted. Fewer South Koreans still believe economic assistance could get the North to abandon its nuclear ambitions.
Yet South Korea is generally regarded as one of Asia's most active democracies.
At the same time, Ms. Park was blamed for stirring up division in the nation's electorate past year when her administration agreed to allow Washington to forge ahead with the deployment of a missile defense shield to South Korea known as the Terminal High Altitude Area Defense (THAAD) system.
Moon Jae-in, the runner-up to Park in 2012, is the favorite to become president and end nine years of conservative rule.
The war between the South and the North never ended in a treaty, and these "tragic" relations, driven by conservative groups, according to the news outlet run by North Korea's ruling party, "revived the foregone period of confrontation and maximized the political and military rivalry between the same race".
The economy is always an issue in every election, Ms Collins said, but it is "highlighted" in this one because of the corruption scandal with Ms Park involved family owned conglomerates, called chaebol, and exposing the underlying corruption of several large corporations in the country. Ahn is closer to the center than Moon, and has been getting support from some conservatives disappointed by Park but who detest Moon.
BBC Korea correspondent Stephen Evans says this could lead to tensions with Washington. "The close cooperation among South Korea, the U.S., China and Japan under the Park government could weaken".