Ukraine crisis: Kerry has 'frank' meeting with Putin
US Secretary of State John Kerry has said he has held a "frank meeting" with President Vladimir Putin during his first visit to Russia since the Ukraine crisis began in early 2014.
Both sides say that eight hours of talks between Mr Kerry and the Russian leadership were a welcome development.
It is the highest-level trip by a US official to Russia since the Ukraine crisis began.
Relations between the two countries deteriorated sharply since then.
The West has criticised Russia for annexing the Crimean peninsula and has accused Russia of arming rebels in eastern Ukraine.
Russia denies arming the rebels or sending troops there - although foreign minister Sergei Lavrov agreed with Mr Kerry at a joint press conference on Tuesday night that the ceasefire in Ukraine is still being violated.
More than 6,000 people have been killed in fighting which began in April 2014 between Ukrainian government forces and Russian-backed rebels in the eastern Donetsk and Luhansk regions.
Mr Kerry said it was critically important that a ceasefire deal agreed in Minsk in February be fully implemented in eastern Ukraine. He said US and EU sanctions against Russia could only be scaled down "if and when" that happens.
"We are in significant agreement on the most important issue of all, which is that [the conflict in Ukraine] will only be resolved by the full implementation of Minsk and all of us have responsibilities to undertake in order to affect that implementation," Mr Kerry said.
The BBC's Bridget Kendall in Sochi says that while there was no concrete sign of progress on any issue, these were the most extensive talks at such a high level between the US and Russia since they fell out over the Ukraine crisis,
Mr Kerry was speaking after holding four hours of talks with Mr Putin - in addition to talks with Mr Lavrov at the Black Sea resort of Sochi. Their negotiations also included Iran, Syria, Libya and Yemen.
He said it was necessary to keep the lines of communication open between the US and Russia and that there was no substitute for direct talks between decision makers. He thanked President Putin for giving so much of his time to explain Russia's positions.
The Russian foreign minister and US secretary of state laid wreaths at a World War Two memorial earlier on Tuesday.
Mr Lavrov said at the joint press conference that the talks had helped Moscow and Washington improve mutual understanding.
Earlier, he said that Russia was ready to co-operate with the US but only on an "equal basis" and without coercion.
A statement by Mr Lavrov on the foreign ministry website said that attempts to pressure Russia through sanctions would only lead to a "dead end".
Analysis: Bridget Kendall, BBC News, Sochi
For President Obama to send his top envoy all the way to Sochi is significant. That the Kremlin has now called the visit positive and said President Putin is ready to discuss a wide range of issues with him is even more important.
One Kremlin adviser said the Russian president had been guided by a fundamental interest in trying to resume normal relations with the Americans.
And both sides emphasised where they shared positions rather than the many points of disagreement.
There was no breakthrough on anything but Mr Kerry pointedly dangled the possibility that Western sanctions could start being rolled back if a peace deal on Ukraine was fully implemented.
Probably an important signal for the Russians, who say they will only co-operate with the US if there is no pressure or coercion.
The visit came on the same day that opposition activists published a report, originally compiled by murdered Kremlin critic Boris Nemtsov, alleging that 220 Russian soldiers had died in two key battles in eastern Ukraine.
Mr Kerry last held talks with Mr Putin in Moscow in May 2013, a meeting for which the Russian leader was three hours late.
The lull in the conflict in eastern Ukraine since February's ceasefire has been punctuated by frequent violations, and on Tuesday Ukraine said three of its soldiers had been killed in the past 24 hours.
The situation in eastern Ukraine
- Reduced but daily clashes, with intense fighting around port city of Mariupol
- Ukrainian government says it has lost full or partial control of 28 towns and villages since 18 February
- Both sides accuse each other of building up weaponry for a new offensive