From London and Paris to New York and Washington, thousands of people took to the streets Saturday in American and European cities to protest United States President Donald Trump's travel ban amid a fierce legal battle over the order.

The biggest demonstration by far took place in the British capital, where an estimated 10,000 people turned out, chanting "Theresa May: Shame on You" to denounce the British prime minister's support for the new US leader.

Brandishing placards declaring "No to scapegoating Muslims" and "Socialism not Trumpism," the protesters moved from the US embassy toward May's Downing Street office.

In an executive order issued on Jan 27, Trump slapped a blanket ban on nationals of seven mainly Muslim countries ─ Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen ─ barring their entry to the United States for 90 days.

Refugees were also barred from entry for 120 days, except those from Syria, who were blocked indefinitely.

On Friday, a US federal judge suspended the ban, a move which the Republican president ─ who took office on Jan 20 ─ condemned and vowed to fight.

Late Saturday, the Justice Department filed its motion to appeal the judge's order, though the legal brief detailing its arguments has yet to come.

"We'll win. For the safety of the country, we'll win," Trump told reporters.

'Dump Trump'

About 3,000 people demonstrated in New York, Trump's hometown where protests against the property magnate-turned-world leader take place almost daily.

Activists and supporters gathered outside the historic Stonewall Inn, a landmark of the gay rights movement in New York's Greenwich Village, to show support for Muslims and others affected by Trump's immigration order.

Democratic Senate minority leader Charles Schumer led the crowd ─ which carried rainbow flags and Americans flags ─ in cheers of "Dump Trump."

In Washington, hundreds marched from the White House to Capitol Hill to show their solidarity and held a peaceful rally across the street from the Supreme Court, decrying the president's recent immigration order.

"Donald, Donald can't you see, we don't want you in DC," chanted the demonstrators in the largely Democratic-leaning US capital.

Many waved homemade signs with slogans like "Love knows no borders" and "Will swap Trump for 1,000 refugees."

"I was born and raised here and for the first time in my life, I don't feel safe," said Abu Bakkar, 26, whose parents are originally from Pakistan.

The Department of Defense consultant said the new president "has revealed hate that's been underground for so long. He has divided one of the greatest countries in the world."

Anti-Trump petition

In Britain, more than 1.8 million people have signed a petition saying Trump should not be afforded a formal state visit because it would embarrass Queen Elizabeth II.

"We're going to bring this capital to a halt on the day he comes over. We are going to make it impossible for him to have a state visit," Chris Nineham, vice-chair of the Stop the War Coalition, told AFP.

The Guardian newspaper said around 10,000 people attended the London protest, while organisers claimed 40,000.

Elsewhere in Europe, about 1,000 people turned out in both Paris and Berlin, while smaller gatherings of several hundred people took place in provincial British cities including Manchester and Birmingham.

On Saturday evening, at least 2,000 people marched near Trump's Florida golf club as the president and First Lady Melania Trump attended a Red Cross gala at his Mar-a-Lago estate.

"You know, my father fought World War II," protestor Rob Resaid said. "For years and years, so many people fought for freedom in this country and now it's being taken away."

"We have to stand now before all of our rights are taken away, before we become the pariah of the world."

Iraqi-American Maryam al-Hassani, 18, told AFP she has family members who were trying to reach America before their travel visas were revoked following Trump's order. An Iraqi flag knotted at her neck, the university student voiced cautious optimism about the court decision lifting the travel ban: "It's important to look at the positives, but at the same time he's still in office for four years."

Victor Veizaga, 68, sported a red, white and blue cap as he rallied with his son and two grandchildren.

"Here is my life and my soul," said Veizaga, who immigrated to the United States from Bolivia in 1970 and has lived in the nation's capital, a Democratic-leaning city where the president's election has been largely unwelcome, for four decades.

Motioning to his grandchildren ─ a boy and a girl who carried neon green signs that read "Love Trumps Hate" and "#FuturePrez" ─ he said the persistent protests have left him feeling hopeful.

"I owe everything to this beautiful country," he said. "We have dignity, we have hope, and we have a future."