Seated next to Putin in Germany, Trump said it was "an honour" to be with Putin. As journalists were briefly allowed in to witness part of the meeting, Trump said that he and Putin had already held "very, very good talks."
"We look forward to a lot of very positive things happening for Russia, for the United States and for everybody concerned," Trump said.
With Secretary of State Rex Tillerson at his side, Trump sat in front of an American flag with his hands clasped together in triangle formation.
Trump offered no details about what issues he and the Russian leader had discussed, describing them only as "various things." Putin was similarly vague, telling reporters through a translator that they were discussing international problems and bilateral issues.
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Still, Putin described the fact that they were meeting as a positive sign in itself, and he said he hoped the meeting would "yield positive results."
"Phone conversations are never enough definitely," Putin said. "If you want to have a positive outcome in bilaterals and be able to resolve most international policy issues, that will really need personal meetings."
Both kept their composure amid the commotion of cameras clicking and journalists lobbying questions as anxious aides moved about nearby. The US leader's son, Donald Trump Jr., took to Twitter to say the noise from the cameras made it difficult to even hear the two leaders' words.
"How many pictures do you need of the same scene?" he said.
Trump's predecessor, President Barack Obama, had notoriously strained ties to Putin, and Trump has expressed an interest in a better US-Russia relationship.
But deep skepticism about Russia in the U.S. And ongoing investigations into whether Trump's campaign coordinated with Moscow during last year's election have made a US-Russia detente politically risky for Trump.
The White House said it has scheduled 35 minutes for the meeting.
In a prelude to their formal sit-down, Trump and Putin shook hands and exchanged broad grins earlier today in a brief exchange caught on video as a leaders' retreat got under way in Hamburg.
In another clip, Trump casually patted Putin on the back as they stood side by side.
Video of the brief exchange was posted to Facebook by the German Cabinet. It was the first known in-person interaction between the two men.
Trump alluded to the campaign controversy as he started the day with a jab at his vanquished Democratic opponent, Hillary Clinton.
He wrote on Twitter that "everyone here is talking" about her campaign chairman's "disgraceful" response after the FBI discovered Democratic Party computers were hacked, a breach later blamed on Russia.
Mrs Trump's office said local police had not cleared her to leave the government guest house where she and Trump were staying because of the protests, in which demonstrators set dozens of cars ablaze.
In the lead-up to the meeting, Trump used a speech in Warsaw yesterday to voice a list of grievances about Russia. He urged Putin's government to "cease its destabilising activities in Ukraine and elsewhere and its support for hostile regimes -- including Syria and Iran -- and to instead join the community of responsible nations in our fight against common enemies and in defense of civilization itself."
In a news conference before he flew to Germany, Trump again refused to unequivocally accept the conclusion by multiple US intelligence agencies that Russia interfered to try to help Trump win last November.
Trump said it could have been Russia, but that other countries could have meddled, too.
"Nobody really knows for sure," Trump said.
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