The talks are seen as a chance to shift relations between the two states
US President Barack Obama has praised Russian PM Vladimir Putin at talks outside Moscow, saying there was an excellent chance to improve ties.
Mr Obama's meeting with the man widely regarded as the most powerful in Russia is taking place on the second day of the American leader's visit to Moscow.
Mr Obama praised Mr Putin for his "extraordinary work" as president and PM as the pair met for the first time.
Mr Putin said Mr Obama's own role would be key in improving relations.
The breakfast meeting was held at Mr Putin's dacha outside the capital.
Richard Galpin, Moscow
In unusually blunt comments last week, Mr Obama described Mr Putin as a man with a cold war mentality who still had one foot in the past.
It provoked the usual sharp response from Mr Putin's office.
The fact that Mr Obama is meeting the prime minister - ostensibly to discuss economic issues - is a reflection of the power Mr Putin still holds.
Many believe he remains the most powerful politician in the country, making the vast majority of all the important decisions.
So if the US president really wants to ensure the relationship between Washington and Moscow is on the mend, he needs to include Mr Putin in his charm offensive.
The two men exchanged pleasantries before their talks, during which Mr Obama himself demonstrated the confusion over Mr Putin's position in the Russian power structure.
"I am aware of not only the extraordinary work that you've done on behalf of the Russian people in your previous role as prime minis-, uh, as president, but in your current role as prime minister."
Mr Putin was made prime minister in 1999 under President Boris Yeltsin, before being elected president the following year. After serving the maximum two terms, he reverted to prime minister in 2008 - though many suspect he still takes the key decisions.
Mr Putin told Mr Obama: "We link hopes for development of our relationship with your name."
Last week, Mr Obama said he thought the former Russian president turned prime minister had "one foot in the old ways of doing business and one foot in the new".
Mr Obama is later due to deliver a major speech on democracy, the global economy and the US-Russian relationship.
On Monday, after meeting Russian President Dmitry Medvedev, he signed eight separate agreements with Russia.
They included a pact to negotiate a new arms control treaty to replace the 1991 Start I pact which expires in December.
A new agreement could see the two countries reducing their nuclear warheads by up to a third, to below 1,700 each within seven years of the treaty's signing.
Separately, Russia also agreed to allow the US military to fly troops and weapons across its territory into Afghanistan, allowing it to avoid using supply routes through Pakistan that are attacked by militants.
However, on the contentious issue of US plans to base parts of a missile defence shield in Eastern Europe, the presidents merely said they had agreed to a joint study into ballistic missile threats and the creation of a data exchange centre.