Two women told the New York Times that the Republican presidential candidate groped or kissed them.
A People magazine reporter also said she was forcibly kissed, while another woman said Mr Trump grabbed her bottom.
Mr Trump’s campaign have rejected the claims and threatened legal action against the NYT.
His camp made public a letter to the US newspaper calling its article “defamatory” and “a politically-motivated effort to defeat Mr Trump’s candidacy”. The NYT said it was standing by its story.
The allegations come less than a week after a video shot in 2005 emerged which showed Mr Trump making obscene remarks about groping women.
Calling the NYT story disturbing, Hillary Clinton’s campaign said it “sadly fits everything we know about the way Donald Trump has treated women”.
What are the allegations?
The New York Times published accounts from two women, Jessica Leeds and Rachel Crooks.
Ms Leeds, now 74, said that when she was 38 she sat next to Mr Trump in the first-class cabin of a flight to New York. Once airborne, he lifted the armrest and began to touch her.
“He was like an octopus… his hands were everywhere,” said Ms Leeds. “It was an assault.”
Rachel Crooks said she was kissed on the lips by Mr Trump outside a lift in Trump Tower when she was a 22-year-old receptionist at a real estate company there.
“It was so inappropriate,” Ms Crooks said. “I was so upset that he thought I was so insignificant that he could do that.”
In People magazine, writer Natasha Stoynoff said an incident took place in December 2005, when she went to interview the Trumps ahead of their first wedding anniversary.
Mr Trump said he wanted to show her around their Florida estate, including one “tremendous” room.
“We walked into that room alone, and Trump shut the door behind us. I turned around, and within seconds he was pushing me against the wall and forcing his tongue down my throat,” Ms Stoynoff wrote.
Another woman, Mindy McGillivray, now 36, told the Palm Beach Post that when she was 23, Mr Trump grabbed her bottom at his Mar-a-Lago club in Florida in 2003.
None of the women reported their accounts to the authorities, but said they shared what happened with friends, colleagues or family.
Other reports that have emerged:
The Guardian quoted an unidentified contestant in the Miss USA 2001 pageant as saying the mogul walked in on her and a fellow contestant while they were naked. “He walked in, he stood and he stared,” she said. Buzzfeed reported a similar claim related to the 1997 Miss Teen USA pageant
According to Yahoo News, Cassandra Searles, who was Miss Washington 2013, in a comment on a Facebook post said Mr Trump “continually grabbed my ass and invited me to his hotel room”
How has Mr Trump reacted?
The Trump camp has not reacted to all the claims, but it came out strongly against the NYT report.
In addition to the legal threat, campaign spokesman Jason Miller accused the paper of launching “a completely false, co-ordinated character assassination”.
Reaching back decades set “a new low for where the media is willing to go in its efforts to determine this election”, he said.
Accounts from Ms Stoynoff and Ms McGillivray were both described by the campaign as lacking “merit or veracity”.
Asked during last Sunday’s debate whether he had kissed or groped women without their consent, Mr Trump said: “No, I have not”, and stressed that he respected women.
How have voters reacted?
The latest polls show Hillary Clinton pulling into a solid lead over her Republican rival.
On Monday, a poll by Reuters/Ipsos gave Mrs Clinton an eight-point lead.
It also showed that a fifth of Republicans thought Mr Trump’s comments about groping woman disqualified him from the presidency.
What about the Republican party?
There has been limited reaction to the newest claims. But since the video emerged, many top Republicans have distanced themselves. The most high-ranking is House Speaker Paul Ryan, whose desertion prompted a furious reaction from Mr Trump.
However, four Republican members of Congress who had urged Mr Trump to step aside immediately after the video was released have since retreated from that position, saying they backed their party. The NYT said they had faced a “fierce backlash” from Trump supporters.