Hateful rhetoric, insulting tweets, or - such as this week since the far-right violence in Charlottesville - his inability to clearly identify and distance himself from right-wing extremism. Many of Trump's statements and attitudes have been met with consternation and outrage the world over.

Trump on Obama: 'Why doesn't he show his birth certificate?'

A black president? For followers of the racist "birther" movement, it was inconceivable. Trump was long the most prominent among them to sow lies about Barack Obama's background, claiming he was born in Kenya, not the US, and was not Christian.

"Why doesn't he show his birth certificate?" Trump said in a 2011 interview for US broadcaster ABC. Trump followed up this initial falsehood during the 2012 presidential campaign, tweeting: "An 'extremely credible source' has called my office and told me that @BarackObama's birth certificate is a fraud."

Trump continued to call Obama's legitimacy into question even after the White House released Obama's birth certificate. Only as a candidate for president himself, in September 2016, did Trump changed course, saying: "President Barack Obama was born in the United States. Period." Neither he nor Obama have publicly said if Trump ever apologized for spreading the lie.

Trump on Mexican immigrants: 'Criminals and rapists'

Mexico and its citizens were a regular target of the Trump campaign. "When Mexico sends its people, they're not sending their best," he said in 2015. "They're sending people that have lots of problems, and they're bringing those problems with us. They're bringing drugs. They're bringing crime. They're rapists." 

"And some," he added, "I assume, are good people."

This view formed the basis for Trump's campaign promise to build a border wall and "make Mexico pay for it."

Trump on dead Muslim soldier's mother: 'Allowed to speak?'

United States Army Captain Humayun Khan was killed in Iraq in 2004. His parents, Americans with Pakistani roots, spoke at the 2016 Democratic National Convention, criticizing Trump's candidacy and its inherent racism. The father, Khizr Khan, spoke; his wife, Ghazala, stood by him.

Trump's interpretation of the scene: "If you look at his wife, she was standing there. She had nothing to say. She probably - maybe she wasn’t allowed to have anything to say."

Many on social media described his comment as disrespectful as well as overtly anti-Muslim.

Trump on Arabs and Muslims: 'Islam hates us' 

Trump's anti-Muslim and Arab sentiments have been routinely quoted. In a 2015 ABC interview, he expressed the completely made-up claim that Arabs celebrated the September 11 attacks. "There were people that were cheering on the other side of New Jersey, where you have large Arab populations. They were cheering as the World Trade Center came down." It never happened.

In early 2016, Trump said on CNN: "I think Islam hates us." And later: "We're having problems with the Muslims, and we're having problems with Muslims coming into the country."

The latter comment led to one of Trump's first major policy moves as president: A travel ban for people from six Muslim-majority countries, the legality of which was challenged and largely struck down in court.

Trump on right-wing extremism: 'Racism is evil,' or is it?

Trump was at first quiet following last weekend's violence between left and right-wing protesters in Charlottesville, Virginia. Then the president blamed "violence on both sides," without explicitly calling out neo-Nazis and other racists. Following fierce criticism at home and abroad, Trump finally made a clear statement: "Racism is evil, and those who cause violence in its name are criminals and thugs."

However, at a press conference just a few days later, Trump changed course: "You had a group on one side that was bad, and you had a group on the other side that was also very violent," he said.

Trump's critics charge that he has not done enough to credibly distance himself from white-supremacist and other right-wing movements. His aide, Steve Bannon, was editor-in-chief of Breitbart News, a mouthpiece of the far-right movement known as alt-right. 

Many right-wing activists and groups feel emboldened by Trump's comments. Critics say he does not do enough to distance himself from white supremacist groups like the Ku Klux Klan or the alt-right movement.

Read more: White supremacy and neo-Nazis in the US - what you need to know