Russian and Ukrainian officials will meet via videoconference on Monday morning in the latest round of talks between the two sides.
And there are signs which offer a glimmer of hope for progress to end the war, which began with Russia's invasion of Ukraine on 24th February.
On Sunday evening, a Russian negotiator reported "significant progress".
"My personal expectation is that this progress will lead very soon to a common position between the two delegations and to documents to be signed" Russian news agencies quotes Leonid Sloutski as saying.
Mykhailo Podoliak, an adviser to Ukrainian President Volodymyr Zelenskyy, tweeted that Moscow had stopped issuing "ultimatums" to Kyiv and started to "carefully listen to our proposals".
"I see the understanding and there is dialogue" Mr Podoliak said, adding later that negotiations were "non-stop in the format of video conferences" and that Monday's session would be to sum up the preliminary results of recent dialogue.
However, Podoliak also said that Ukraine's positions had not changed: "peace, an immediate ceasefire, the withdrawal of all Russian troops, and only then can we talk about our neighborly relations and our political differences" he tweeted.
Optimistic statements as fighting continues
Over the weekend the Ukrainian and Russian presidents made statements that seemed more optimistic in tone.
Vladimir Putin referring to "positive progress" in the talks on Friday and Mr Zelenskyy said on Saturday there had been a "fundamentally different approach" from Moscow towards discussions.
And on Sunday Zelenskyy said that his delegation had "a clear task: to do everything to ensure a meeting of the presidents. A meeting that people are waiting for, I'm sure".
Monday morning's talks get underway as fierce fighting continues across Ukraine, spreading to the west of the country.
The British Ministry of Defence says Russian naval forces have "established a remote blockade of Ukraine's Black Sea coast, effectively isolating Ukraine from international maritime trade".
The port city of Mykolaiv was again targeted by shelling on Sunday, killing nine people, authorities said.
Further east the city of Mariupol is under siege, waiting the arrival of a humanitarian aid convoy. An adviser to the mayor, Petro Andryushchenko, said on Sunday evening that vehicles had to turn around because of ongoing Russian fire.
In the north, Kyiv is now "a city under siege", in the words of an adviser to the Ukrainian president. Its inhabitants have stocked up on food and medicine and the authorities have set up barricades as Russian forces advance.
President Zelenskyy visited wounded Ukrainian military personnel at a Kyiv hospital on Sunday, shaking hands with them and posing for selfies.
During the night from Sunday to Monday, the Ukrainian Air Force announced that Russian planes were trying to bomb Ukrainian defensive positions in the Kyiv region.