US President Donald Trump has met Pope Francis at the Vatican and appeared to be moved by the occasion, telling the pontiff that he "won't forget what you said".

Mr Trump, who is midway through his nine-day maiden international journey, held a private 30-minute meeting with the pontiff which was laden with religious symbolism and ancient protocol.

The president, who was accompanied by his wife and several aides, greeted Francis in a room on the second floor of Apostolic Palace.

The men shook hands and Mr Trump could be heard thanking the pope and saying it was "a great honour" to be there.

They then posed for photographs and then sat down at the papal desk, the pope unsmiling, as their private meeting began.

It ended half an hour later when Francis rang the bell in his private study.

The pontiff was then introduced to members of Mr Trump's delegation, including his wife Melania, his daughter Ivanka and son-in-law Jared Kushner, as well as aides Hope Hicks and Dan Scavino. As is tradition, the pope and president then exchanged gifts.

The meeting could provide powerful imagery to Catholic voters back in the United States as well as the possibility for conflict between a president and a pope who have not often seen eye-to-eye.

The two men's often opposite world views collided head-on early last year, when Francis was sharply critical of Mr Trump's campaign pledge to build an impenetrable wall on the Mexican border and his declaration that the United States should turn away Muslim immigrants and refugees.

"A person who thinks only about building walls, wherever they may be, and not building bridges, is not Christian," Francis said then.

The pontiff has been a vocal advocate for aiding refugees, particularly those fleeing the violence in Syria, deeming it both a "moral imperative" and "Christian duty" to help.

Mr Trump has never been one to let an insult, perceived or real, go by without a response, and he made no exception for the world's best-known religious leader.

He called Francis "disgraceful" for doubting his faith.

And even the pontiff's congratulatory message sent to mark Mr Trump's inauguration contained a sly reference to their disagreement, as the pope wrote that he hoped the United States' international stature would "continue to be measured above all by its concern for the poor, the outcast and those in need".

Mr Trump arrived in Rome on Tuesday evening, his motorcade closing a busy Italian motorway just after rush hour and prompting hundreds of onlookers to briefly step out of their gridlocked cars to look at the fleet of armoured vehicles.

He spent the night at the US ambassador to Italy's residence.

Though both Mr Trump and Francis are known for their unpredictability, papal visits with heads of state are carefully arranged bits of political and religious theatre that follow a specific programme, with little room for deviation or unwanted surprises.

After the meeting, the pope gave the president copies of his three main teaching documents as parting gifts, as he typically does for visiting heads of state.

The red leather-bound booklets to some degree define his papacy and priorities.

Some of the main themes contained in them contrast sharply with Mr Trump's policies and campaign promises, particularly concerning approaches to the environment and income inequality.

Mr Trump's gift for Francis was wrapped in a big blue box.

The president said he was delivering "books from Martin Luther King. I think you'll enjoy them. I hope you do".

The pope's gifts to Mr Trump also included a medal by a Roman artist depicting an olive, which is a symbol of peace.

The president responded: "We can use peace."