BBC
Mr Kim took centre stage at a memorial service in Pyongyang's main square a day after his father's funeral.
Kim Yong-nam, formally the number two leader, told a million-strong crowd their sorrow would be turned into strength "1,000 times greater under the leadership of comrade Kim Jong-un".
State TV showed Kim Jong-un surrounded by top government and army officials.
The memorial event appeared to be the Kim dynasty's unofficial handover of power, says the BBC's Lucy Williamson in neighbouring South Korea.
A three-minute silence was also held, after which trains and ships throughout the country sounded their horns.
Kim Jong-il died of a heart attack on 17 December, aged 69, state media said. He had ruled North Korea since the death of his father Kim Il-sung in 1994.
'Military first'
"Respected Comrade Kim Jong-un is our party, military and country's supreme leader who inherits great comrade Kim Jong-il's ideology, leadership, character, virtues, grit and courage," Kim Yong-nam told the massive crowd gathered in Kim Il-sung square.
"The fact that he completely resolved the succession matter is Great Comrade Kim Jong-il's most noble achievement."
A top military official, Kim Jong-gak, also addressed the crowd.
"Our people's military will serve comrade Kim Jong-un at the head of our revolutionary troops and will continue to maintain and complete the Songun accomplishments of great leader Kim Jong-il," he said.
Songun refers to the "military-first" policy - channelling funds into the military.
Half-mast
On Wednesday, thousands stood weeping and wailing in the snow as Kim Jong-il's funeral cortege passed, images from state television showed.
Kim Jong-un - Mr Kim's third son - cried as he walked alongside the hearse. Tens of thousands of soldiers lined up to bow their heads in homage in the city's main square.
Kim Jong-un - who is thought to be in his late 20s and who has little political experience - was accompanied by his uncle, Chang Song-taek.
Mr Chang is expected to be a key player as the younger Kim consolidates power.
Kim Jong-il - known in North Korea as the "Dear Leader" - was in the process of formalising Kim Jong-un as his successor when he died.
However, the transition was not complete, leaving regional neighbours fearful of a power struggle in the nuclear-armed pariah state.
Mr Kim's two older sons, Kim Jong-nam and Kim Jong-chol, were not seen at the funeral.
No foreign delegations have attended any of the events. However, UN offices around the world lowered their flags to half-mast.
A spokesman at the UN headquarters in New York said that the move had been requested by Pyongyang's UN mission but was part of normal protocol for the funeral of any head of state.
Many thousands of North Koreans took part in Thursday's choreographed memorial
North Korea has hailed late leader Kim Jong-il's son, Kim Jong-un, as "supreme leader of the party, state and army".
Kim Yong-nam, formally the number two leader, told a million-strong crowd their sorrow would be turned into strength "1,000 times greater under the leadership of comrade Kim Jong-un".
State TV showed Kim Jong-un surrounded by top government and army officials.
The memorial event appeared to be the Kim dynasty's unofficial handover of power, says the BBC's Lucy Williamson in neighbouring South Korea.
A three-minute silence was also held, after which trains and ships throughout the country sounded their horns.
Kim Jong-il died of a heart attack on 17 December, aged 69, state media said. He had ruled North Korea since the death of his father Kim Il-sung in 1994.
'Military first'
"Respected Comrade Kim Jong-un is our party, military and country's supreme leader who inherits great comrade Kim Jong-il's ideology, leadership, character, virtues, grit and courage," Kim Yong-nam told the massive crowd gathered in Kim Il-sung square.
"The fact that he completely resolved the succession matter is Great Comrade Kim Jong-il's most noble achievement."
Kim Jong-un (in buttoned black overcoat) bows at the memorial in Pyongyang
"Our people's military will serve comrade Kim Jong-un at the head of our revolutionary troops and will continue to maintain and complete the Songun accomplishments of great leader Kim Jong-il," he said.
Songun refers to the "military-first" policy - channelling funds into the military.
Half-mast
On Wednesday, thousands stood weeping and wailing in the snow as Kim Jong-il's funeral cortege passed, images from state television showed.
Continue reading the main story
Before him a vast concourse of the North Korean people. They filled Kim Il-sung square, hundreds of thousands, perhaps a million. Soldiers and civilians lined up in silent, arrow-straight rows.
Kim Jong-un was hailed as the "supreme leader" of the party, state and army, who has inherited the character and ideology of his father. He did not address the crowd himself.
Marshalling the population on such an epic scale is a display by North Korea's rulers that their grip on power remains firm.
But Kim Jong-un is not yet 30. North Korea's neighbours fear infighting at the top, or that the young, untried and untested new leader could launch military attacks to cement his rule.
Analysis
Kim Jong-un stood impassively on a balcony above a portrait of his father, flanked by military commanders and senior members of the party.Before him a vast concourse of the North Korean people. They filled Kim Il-sung square, hundreds of thousands, perhaps a million. Soldiers and civilians lined up in silent, arrow-straight rows.
Kim Jong-un was hailed as the "supreme leader" of the party, state and army, who has inherited the character and ideology of his father. He did not address the crowd himself.
Marshalling the population on such an epic scale is a display by North Korea's rulers that their grip on power remains firm.
But Kim Jong-un is not yet 30. North Korea's neighbours fear infighting at the top, or that the young, untried and untested new leader could launch military attacks to cement his rule.
Correspondents say the ceremonies echoed the displays of pomp and military might that marked the death of Kim Il-sung, in 1994.
Kim Jong-un - who is thought to be in his late 20s and who has little political experience - was accompanied by his uncle, Chang Song-taek.
Mr Chang is expected to be a key player as the younger Kim consolidates power.
Kim Jong-il - known in North Korea as the "Dear Leader" - was in the process of formalising Kim Jong-un as his successor when he died.
However, the transition was not complete, leaving regional neighbours fearful of a power struggle in the nuclear-armed pariah state.
Mr Kim's two older sons, Kim Jong-nam and Kim Jong-chol, were not seen at the funeral.
No foreign delegations have attended any of the events. However, UN offices around the world lowered their flags to half-mast.
A spokesman at the UN headquarters in New York said that the move had been requested by Pyongyang's UN mission but was part of normal protocol for the funeral of any head of state.
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