It's an honour to be with you," Mr Trump told Mr Putin, who responded: "I'm delighted to meet you personally."
The US and Russian leaders say they want to repair ties damaged by crises including Russia's alleged meddling in the US election.
Climate change and trade are set to dominate the two-day G20 meeting.
Violent clashes between protesters and police are taking place in the streets outside the venue, with dozens left injured.
A huge police operation is trying to keep demonstrators - who are protesting against the presence of Mr Trump and Mr Putin, climate change and global wealth inequalities - well away from the summit venue, and water cannon have been deployed.
TRUMP AND PUTIN MEET FACE TO FACE IN FIRST MEETING
US First Lady Melania Trump has been unable to leave her hotel in Hamburg because of the protests.
"Putin and I have been discussing various things, and I think it's going very well," Mr Trump told reporters while sitting alongside Mr Putin at the start of the talks, which were open to the media.
"We've had some very, very good talks. We're going to have a talk now and obviously that will continue. We look forward to a lot of very positive things happening for Russia, for the United States and for everybody concerned."
Mr Putin, via a translator, said that while they had spoken by phone, a phone conversation was never as good as meeting face to face.
Both men ignored shouted questions from reporters as the meeting went into private session.