For example, Pope Francis gave President Trump a large medallion depicting an olive branch as a symbol of peace. President Donald Trump and his wife Melania listen to a guide during their visit to the Sistine Chapel and the Vatican Museums, at the Vatican, Wednesday, May 24, 2017.
"What do you feed him on?"
In Saudi Arabia, the first lady dressed conservatively for her arrival Saturday in the capital of Riyadh.
The first lady laughed and said "Yes". But her staff confirmed later that she had appreciated Francis had produced a custom-made joke for her benefit.
We don't know exactly what the two men discussed during their 30-minute, one-on-one meeting behind closed doors.
Part of Trump's motivation for meeting the pope was to dramatize how the three major religions should rally against the threat from Islamist militants.
Like Trump, the Argentine pope has shown a predilection for unscripted comments that have shaken the staid Vatican bureaucracy, as when he criticized candidate Trump's proposed stricter immigration policies including a border wall as "not Christian".
"We thought that this trip was essential to put together the Muslim faith, the Jewish faith and then the Catholic faith, the Christian faith", said a senior White House official who briefed reporters on Trump's Air Force One flight to Rome.
That makes him overweight but not obese in the most common criterion used by doctors - but observers are sceptical, believing his true weight does take him into obese territory.
Trump later tweeted that it had been an "honor of a lifetime" to meet the pope.
Ivanka Trump also dressed modestly, and left her head uncovered.
Trump also presented the pope with a small piece of granite from the Martin Luther King Jr. memorial in Washington, D.C., and a bronze sculpture titled Rising Above, by the American sculptor Geoffrey Smith, representing "hope for a peaceful tomorrow", according to the White House.
After visiting with His Holiness, Trump went on to meet with Italian President Sergio Mattarella and Italian Prime Minister Paolo Gentiloni.
A Vatican statement on the meeting highlighted "the joint commitment in favour of life, and freedom of worship and conscience". The president was heard thanking the pope and saying it was "a great honor" to be there.
Every woman in the US delegation wore a veil, including Ivanka Trump, the president's daughter who converted to Judaism before marriage.
The president's prior anti-Muslim rhetoric - including his musing that Islam "hates" the West - is the antithesis of what the pope has been preaching about a need for dialogue with Muslims. But the timing on Wednesday was tight as Francis had his weekly Wednesday general audience, the thousands of pilgrims on hand forced Trump's motorcade to enter Vatican City from a side entrance rather than the grand entrance through St. Peter's Square. Trump has said climate change might be a hoax perpetrated by the Chinese.
Trump's visit to the Eternal City follows two stops in the Middle East where he visited the cradles of Islam and Judaism.