Many of them shouted the slogan “Not my president”. Others burned orange-haired effigies of the businessman.
Donald Trump will become the 45th president after securing a surprise victory over Hillary Clinton.
The president-elect is due to meet current White House incumbent Barack Obama for talks aimed at ensuring a smooth transition.
President Barack Obama – who had branded Donald Trump “unfit” for office and campaigned against him – urged all Americans to accept the result of Tuesday’s election.
“We are now all rooting for his success in uniting and leading the country,” he said.
Hillary Clinton also told supporters Donald Trump had to be given a “chance to lead”.
Despite their calls, protesters gathered in several cities across the country.
In New York, thousands marched on Trump Tower, attacking Donald Trump’s policies on immigration, gay rights and reproductive rights. Fifteen people were arrested, the New York Times reported.
Protests were largely peaceful but in Oakland, California, some demonstrators smashed shop windows and threw missiles at riot police, who reportedly responded with tear gas.
A mass anti-Trump rally shut down the key 101 freeway in Los Angeles.
In Chicago, crowds blocked the entrance to Trump Tower, chanting: “No Trump, No KKK, No Fascists USA”.
In Portland, Oregon, demonstrators temporarily closed an interstate highway.
In Washington DC, protesters held a candlelit vigil. Organizer Ben Wikler told the crowd: “We are here because in these darkest moments, we are not alone.”
Demonstrations also took place in Philadelphia, Boston, Seattle and San Francisco, among other cities.
In his victory speech, Donald Trump vowed to “bind the wounds of division”, after an acrimonious election contest, and to be “president for all Americans”.
White House spokesman Josh Earnest has insisted Barack Obama will be sincere about ensuring a smooth handover when he meets Donald Trump, although he added: “I’m not saying it’s going to be an easy meeting.”
Donald Trump will be accompanied to the White House by his wife, Melania, who will have a meeting with First Lady Michelle Obama in the White House residence.
The president-elect’s transition team for the 10-week period until inauguration will be led by New Jersey Governor Chris Christie.
Donald Trump, who has never held elected office, has said his immediate priorities will be restoring the country’s infrastructure and doubling its economic growth.
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