Ukrainian President Volodymyr Zelensky has said Ukraine is moving "very carefully" after Russia announced it was pulling out of the southern Ukrainian city of Kherson.
"The enemy does not give us gifts, does not make 'goodwill gestures', we win it all," Mr Zelensky said in his daily address.
"Therefore, we are moving very carefully, without emotions, without unnecessary risk, in the interests of liberating all our land and so that the losses are as small as possible."
Earlier, Russian Defence Minister Sergei Shoigu ordered his troops to withdraw from the west bank of the Dnipro River in the face of Ukrainian attacks near the southern city of Kherson, a significant retreat and potential turning point in the war.
Officials in Ukraine and civilians have reacted with caution to the announcement.
An adviser to Mr Zelensky - Mykhailo Podolyak - said some Russian forces were still in Kherson and it was too early to talk of a withdrawal.
"Until the Ukrainian flag is flying over Kherson, it makes no sense to talk about a Russian withdrawal," he said in a statement.
On Twitter, he said: "We see no signs that Russia is leaving Kherson without a fight.
"Ukraine is liberating territories based on intelligence data, not staged TV statements."
Kherson city was the only regional capital Russia had captured since its invasion in February, and its abandonment would be a major setback for what Moscow terms its "special military operation" in Ukraine.
Regaining the Kherson region would return important access to the Sea of Azov to Ukraine.
In televised comments, Russian General Sergei Surovikin, in overall command of the war, said it was no longer possible to supply Kherson city. He said he proposed to take up defensive lines on the eastern bank of the river.
Mr Shoigu responded: "I agree with your conclusions and proposals. Proceed with the withdrawal of troops and take all measures to transfer forces across the river."
The news followed weeks of Ukrainian advances towards the city and a race by Russia to relocate tens of thousands of its residents.
"We will save the lives of our soldiers and fighting capacity of our units. Keeping them on the right (western) bank is futile. Some of them can be used on other fronts," General Surovikin said.
In recent weeks there has been speculation that Moscow could either withdraw its forces from the west bank of the Dnipro or dig in for a bloody battle in the coming days or weeks.
Earlier, the main bridge on a road out of Kherson city was blown up.
Photos showed the span of the Darivka bridge on the main highway east out of Kherson completely collapsed into the water of the Inhulets River, a tributary of the Dnipro River.
Ukrainians who posted photos of the destroyed bridge over the Inhulets today speculated that it had been blown up by Russian troops in preparation for a retreat.
Vitaly Kim, the Ukrainian governor of Mykolaiv region, which borders Kherson, suggested Ukrainian forces had pushed some Russians out.
"Russian troops are complaining that they have already been thrown out of there," Mr Kim said in a statement on his Telegram channel.
The pull-out announcement had been anticipated by Russia's influential war bloggers, who described it as a bitter blow.
"Apparently we will leave the city, no matter how painful it is to write about it now," said the War Gonzo blog, which has more than 1.3 million subscribers on Telegram.
"In simple terms, Kherson can't be held with bare hands," it said. "Yes, this is a black page in the history of the Russian army. Of the Russian state. A tragic page."
Kherson city controls both the only land route to the Crimean peninsula and the mouth of the Dnipro, the river that bisects Ukraine.
It comes as senior United Nations officials plan to meet members of a high-level Russian delegation in Geneva on Friday to discuss the Ukraine grain deal, a UN spokesperson said.
"They will continue ongoing consultations in support of the efforts by the Secretary-General Antonio Guterres on the full implementation of the two agreements signed on 22 July in Istanbul," the spokesperson said in a statement.
Meanwhile the Kremlin said that relations between Moscow and Washington would "remain bad" after the US midterm elections.
US President Joe Biden, who has been a key ally to Kyiv and provided weapons and financial backing, could be constrained in his support for Ukraine if Republicans win majorities in the Senate and the House of Representatives.
But NATO chief Jens Stoltenberg denied that would undermine Western military backing for Ukraine.
After talks with UK Prime Minister Rishi Sunak, Mr Stoltenberg said: "It's absolutely clear that there's strong bipartisan support in the United States for a continued support for Ukraine, and that's not changed."
It comes after it was confirmed by the Foreign Office that a British man has died fighting in Ukraine.
"We are supporting the family of a British national who has lost his life in Ukraine and we are in touch with the local authorities in connection with his death," the spokesperson told AFP in a statement.
The man's son named him as Simon Lingard, who died in Ukraine on 7 November.
Fresh wave of refugees expected
Eastern European countries are preparing for a possible wave of Ukrainian refugees as Russia targets power and heating plants ahead of winter, with President Zelensky saying about four million people are already without power.
Mr Zelensky said 14 regions plus the capital Kyiv were without power and Ukraine's electrical grid operator Ukrenergo said scheduled hourly power outages would affect the whole of the country.
Russian forces have targeted Ukraine's energy infrastructure with missiles and drone strikes in the run up to winter, when mean temperatures typically drop to several degrees below zero Celsius, with lows of -20C.
Some 6.9 million people are believed displaced internally within Ukraine and east European countries such as Slovakia and Hungary are preparing for an influx in coming months.
"An increase in numbers is being felt and is expected. It is currently up 15%," said Roman Dohovic, an aid coordinator for the eastern Slovak city of Kosice.
Ukrainian forces have been on the offensive in recent months while Russia had been regrouping to defend areas of Ukraine it still occupies, having called up hundreds of thousands of reservists over the past month.
Mr Zelensky said his forces would not yield "a single centimetre" in battles for the eastern Ukrainian region of Donetsk, while Russian-installed officials said Ukrainian forces were moving into a southern town with tanks.
The focal points of the conflict in the industrial region of Donetsk are around the towns of Bakhmut, Soledar and Avdiivka, which have seen the heaviest fighting since Russian forces invaded Ukraine in late February.
"The activity of the occupiers remains at an extremely high level - dozens of attacks every day," Mr Zelensky said in his nightly video address last night.
"They are suffering extraordinarily high losses. But the order remains the same - to advance on the administrative boundary of Donetsk region. We will not yield a single centimetre of our land," he said.
The region is one of four Russia said it annexed in September. Fighting had been going on there between Ukrainian military and Russian proxy forces since 2014, the same year Russia annexed Crimea in the south.
More on Russian invasion of Ukraine
Kyiv-based military analyst Oleh Zhdanov said yesterday that 21 Russian conscripts had surrendered to Ukrainian forces around Svatove in Luhansk region.
"These poor mobilised men - really poor, they had had nothing to eat or drink in three days - of course they decided to surrender," Mr Zhdanov said on his YouTube channel.
Fierce fighting in southern town
A Russian-installed mayor in the town of Snihurivka, east of the southern city of Mykolaiv, was cited by Russia's RIA news agency as saying residents had seen tanks and that fierce fighting was going on.
"They got into contact during the day and said there were tanks moving around and, according to their information, heavy fighting on the edge of the town," the mayor, Yuri Barabashov, said, referring to the residents.
"People saw this equipment moving through the streets in the town centre," he said.
Kirill Stremousov, deputy head of the Russian-installed administration in the Kherson region, said on the Telegram messaging service that Ukrainian forces had tried to advance on three fronts, including Snihurivka.
Reuters was unable to verify the battlefield reports. There was no official word on the situation in the town from military officials in either Ukraine or Russia.
Giving an update on the situation in the neighbouring southern region of Kherson, the Ukrainian military yesterday evening accused Russian troops of more looting and destruction of infrastructure.
The UN General Assembly is due to vote next week on a draft resolution recognising that Russia must be responsible for reparation in Ukraine for the injury, including any damage, caused by "internationally wrongful acts".
The text has been put forward by Ukraine, Canada, Guatemala and the Netherlands.
Three-quarters of the 193-member General Assembly denounced Russia's invasion in a vote in March, and in October condemned its self-proclaimed annexation of parts of Ukraine.
