The visa-restricted countries are Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen.
Here are some of the cases known about so far, of people stopped either before boarding flights to the United States or on arrival there.
An Iraqi family was barred from flying from Cairo to New York on Saturday. Egyptian officials said the family had visas and were in transit when they were prevented from boarding the EgyptAir plane to New York’s JFK airport.
Fuad Sharef, 51, and his wife and three children were instead forced to board a flight back to Arbil in Iraq’s autonomous region of Kurdistan, he told AFP.
“I had sold my house, my car, my furniture. I resigned from work and so did my wife. I took my children out of school,” said Sharef of the family’s preparations to resettle in Nashville, Tennessee, under a special immigrant visa.
“Donald Trump destroyed my life. My family’s life. I used to think America was a state of institutions but it’s as though it’s a dictatorship,” he said.
At least three Iranian nationals were stopped at Vienna airport on Saturday as they sought to travel to the US, a spokesman for Austrian Airlines said.
The trio, an elderly couple and a young woman who all came from Isfahan, had valid US visas, Peter Thier told the Austria Press Agency.
The couple remained stranded in the Austrian capital for the night while the woman had flown back to Iran, he added.
Vienna is an important transit hub for Iranians travelling to the US so there could be more cases.
The New York Times reported that two Iraqi refugees who landed at New York’s John F. Kennedy airport were detained by authorities late Friday, hours after Trump signed the tough new measures.
The American Civil Liberties Union and other advocacy groups filed a legal challenge to Trump’s order. Lawyers representing the pair also filed a court appeal for their release, saying they were being unlawfully detained.
Shortly after midday Saturday one of them was released, the newspaper said, naming him as Hameed Khalid Darweesh, an interpreter who worked on behalf of the US government in Iraq.
The paper said he cried as he spoke to reporters, miming being handcuffed behind his back. “What I do for this country? They put the cuffs on,” it quoted him as saying.
The other Iraqi remained in custody.
Two travel agencies told AFP they had been instructed by Etihad Airways, Emirates and Turkish Airlines not to sell US tickets or allow Iranians holding American visas to board US-bound flights.
An Iranian studying in California who was visiting home said Saturday that she could not return because her ticket had been cancelled under the new restrictions.
“I had a ticket for Turkish Airlines on February 4, but it has been cancelled,” the girl who did not wish to be identified told AFP.
Iran denounced the US ban as “insulting” and said it will ban Americans from entering the country. The Islamic state’s foreign ministry called the decision “illegal, illogical and contrary to international rules”.
Dutch flag carrier KLM said it had stopped seven passengers from boarding its flights, five of whom it had been able to contact before departure.
The identities of the seven, their countries of origin and destinations were not given.
“We would have liked to have had them fly with us, but it would not have made much sense because they would have been denied entry” to the United States, KLM spokesman Manel Vrijenhoek told AFP.