Barack Obama says he is now within reach of winning the Democratic Party's nomination to run in the US presidential election in November.
Senator Obama is set to win a majority of "pledged" delegates, who are won in primary elections, although he is still short of a confirmed overall victory.
US media have projected a win for him in Tuesday's Oregon primary contest.
But his rival Hillary Clinton won Kentucky's primary by a large margin, and has vowed to fight on.
Mr Obama chose to address his supporters on Tuesday night in Iowa, scene of this year's first presidential selection contest.
"We have returned to Iowa with a majority of delegates elected by the American people," he told his supporters.
"You have put us within reach of the Democratic nomination for president of the United States," he added.
He also congratulated his opponent, saying Senator Clinton "has shattered myths and broken barriers and changed the America in which my daughters and yours will come of age".
In the primaries so far, Mr Obama has secured more of the delegates who will choose the party's nominee at the Democratic National Convention.
Correspondents say he will be hoping that his newly-secured majority of pledged delegates will persuade the remaining undecided "super-delegates" - party elders who get an automatic vote at the convention - to rally behind him.
Hailing her Kentucky victory, Mrs Clinton said she would fight "until we have a nominee - whoever she may be".
And in a passage seemingly directed at the remaining undecided super-delegates, Mrs Clinton argued that she would be "best positioned to win in November" against Republican candidate Senator John McCain.
Mrs Clinton also said she was "winning the popular vote" over Mr Obama.
The claim has been questioned by the Obama campaign, which argues that Mrs Clinton's popular vote calculation includes disputed contests in Michigan and Florida, and does not include results from several caucus states in which Mr Obama did well.
So far, Mr Obama has won 1,956 of the 2,026 delegates he needs to clinch the nomination.
BBC North America editor Justin Webb says Mr Obama wants to suggest without quite saying so that the primary season is closing and the general election race is beginning.
Mrs Clinton has warned Mr Obama against declaring premature victory, saying it would be a "slap in the face" to her millions of supporters so far and those states yet to vote.
In Kentucky, Mrs Clinton won 65% of the vote to Mr Obama's 30%.
In Oregon, with more than half of the state's precincts reporting, Mr Obama was leading Mrs Clinton by 16 points.
Exit polls conducted by the Associated Press news agency suggested that Mrs Clinton's Kentucky victory was rooted in her strong support among the state's many white working-class voters.
Three-quarters of white voters without a college education backed the New York senator, the poll suggested.
Racial attitudes may also have motivated some Kentucky voters' decisions: of the 20% of white voters who said that race a had played a part in their choice, nine out of 10 backed Mrs Clinton.
Phone surveys of Oregon voters painted a strikingly different picture of that state's electorate.
Mr Obama won 60% of the state's white voters, and even blue-collar white voters were evenly divided between the two candidates.
Just 10% of the state's voters said that race was important when deciding their vote, and they were split evenly between Mr Obama and Mrs Clinton.
As the results were coming in, Mr Obama, Mrs Clinton and Mr McCain revealed details of how much money they had raised in April.
Mr Obama raised $31m (£15.5m), less than he raised in March, but still the highest of the three presidential hopefuls.
Mrs Clinton's campaign took in $22m (£11.5m), its second best fundraising month so far in the campaign, while Mr McCain managed to raise $18m (£9m), his best ever monthly total.
