Ukraine’s military claims to have broken through Russia’s first line of defence in the occupied Kherson region.
The reported push appears to form part of a long-awaited counter-offensive being launched by Kyiv in an attempt to retake the country’s south.
It follows weeks of Ukrainian attacks aimed at cutting off Russian forces there from main supply routes.
Russia’s military claims that Ukrainian troops suffered “heavy losses” during an unsuccessful attacking attempt.
The claims by both Ukraine and Russia have not been independently verified.
Russia has occupied large swathes of Ukraine’s Kherson region since its invasion began on 24 February.
Early on Monday, Ukraine’s Kakhovka operational group in the south said that one regiment of Russian-backed forces had left its positions in the Kherson region. It added that Russian paratroopers providing the back-up had fled the battlefield.
Oleksiy Arestovych, adviser to President Volodymyr Zelensky’s head of office, later also said that Ukraine’s armed forces “have broken through the frontline in several places”.
Meanwhile, eyewitnesses reported hearing more blasts in the cities of Kherson and Nova Kakhovka, about 55km (34 miles) north-east from the regional capital. The key crossings across the Dnipro River in the two locations have been repeatedly targeted by Ukraine’s military in recent weeks.
Russia’s state-run Ria Novosti news agency reported that Nova Kakhovka was left without electricity and water supply overnight.
In his late night video address, President Zelensky issued a stark warning to Russian forces: “If they want to survive, it is time for Russian soldiers to flee. Go home.”
Mr Zelensky and other top Ukrainian officials have been tight-lipped about details of the reported counter-offensive, urging Ukrainians to be patient.
Responding to the Ukrainian claims, Russia’s defence ministry said that Ukrainian troops had attempted an offensive in the Kherson and neighbouring Mykolaiv regions.
The ministry is quoted by Russia’s state-run news agencies as saying this operation had failed, and that the Ukrainian troops had “suffered heavy losses”.
Kyiv officials claim to have used US-supplied Himars rocket systems to destroy three bridges crossing the Dnipro River.
According to Western military sources, Kyiv’s strikes are part of a targeted effort to cut off Russian troops on the right (western) bank of the river with the ultimate goal of recapturing the entire Kherson region.
Moscow has relied on these bridges to resupply their troops.
A deadlock that could break
Analysis by the BBC’s Hugo Bachega in Kyiv
Ukraine has long been expected to launch a major offensive to retake Kherson. We could be seeing the beginning of it, although any operation is unlikely to be easy.
Kherson has been under occupation since the early days of the war, and it’s one of the largest Ukrainian cities in Russian hands.
For weeks, Ukrainian forces have repeatedly targeted Russian positions deep inside captured territory, away from the front lines.
It’s only been possible because of the sophisticated weapons supplied by the West – and it’s having a destabilising effect on the invading forces.
The conflict seems to be at a deadlock, with neither side making significant gains. This could be about to change.
Russia captured the city of Kherson and its surrounding region with relatively little resistance in the early days of the invasion.
The city, which had a population of 290,000 before the war, is the only regional capital to have been taken by Russian forces and is currently administered by Moscow-backed officials.
According to Russia’s Tass news agency, officials in the Kherson region have started moving forward with plans to hold a referendum on formally joining Russia, prompting accusations by the US that Russia could be preparing to illegally annex parts of occupied southern Ukraine.
Last month, Russia said its military focus was no longer only on eastern Ukraine but on its southern regions of Kherson and Zaporizhzhia too.
In a separate development on Monday, Russian-installed officials in the Zaporizhzhia region claimed that a Ukrainian missile strike punched a hole in the roof of a fuel depot at the Zaporizhzhia nuclear power plant.
The claim has not been independently verified.
In recent weeks, both Ukraine and Russia have accused each other of shelling Europe’s biggest nuclear station, which was seized by Russia in early March. Moscow has kept Ukrainian personnel to operate the station.
An inspection team from the UN nuclear watchdog is expected to arrive at the plant later this week, the organisation’s head says.