Ukrainian forces have reported heavy Russian shelling and attempts to advance on several towns in the eastern region of Donetsk.
The area has become a key focus of the near six-month war, but forces say they had repelled many of the attacks.
The General Staff of Ukraine's armed forces have also reported Russian shelling of more than a dozen towns on the southern front - particularly the Kherson region, mainly controlled by Russian forces, but where Ukrainian troops are steadily capturing territory.
Much attention has been focused on the Zaporizhzhia nuclear plant in southern Ukraine amid fears of a catastrophe over renewed shelling in recent days, which both Russia and Ukraine have blamed on each other.
United Nations Secretary-General Antonio Guterres has called for the establishment of a demilitarised zone and Ukrainian President Volodymyr Zelensky has warned Russian soldiers who shoot at Europe's largest nuclear power station or use it as a base to shoot from, that they will become a "special target" of Ukrainian forces.
The Zaporizhzhia plant dominates the south bank of a vast reservoir on the Dnipro River.
Ukrainian forces controlling the towns and cities on the opposite bank have come under intense bombardment from the Russian-held side.
The International Atomic Energy Agency, which seeks to inspect the plant, has warned of a nuclear disaster unless fighting stops. Nuclear experts fear fighting might damage the plant's spent fuel pools or reactors.
A spokeswoman for Russia's foreign ministry said it would do all it could to allow IAEA specialists to visit the plant.
"In close cooperation with the agency and its leadership, we will do everything necessary for the IAEA specialists to be at the station and give a truthful assessment of the destructive actions of the Ukrainian side," spokeswoman Maria Zakharova said.
Mr Zelensky said Ukraine had many times proposed different formats to the Russian leadership for peace talks, without progress.
"So we have to defend ourselves, we have to answer every form of terror, every instance of shelling - the fierce shelling which does not let up for a single day," he said in video remarks late last night.
Fighting in east and south
Ukraine, where parliament today extended martial law for a further three months, has said for weeks that it is planning a counteroffensive to recapture Zaporizhzhia and neighbouring Kherson province, the largest part of the territory Russia seized after its invasion started and still holds.
Ukraine's military command said early yesterday that Russian soldiers had continued unsuccessfully to attack Ukrainian positions near Avdiivka, which, since 2014, has become one of the outposts of Ukrainian forces near Donetsk.
Ukrainian military expert Oleg Zhdanov said the situation was particularly difficult in Avdiivka and nearby towns, such as Pisky.
"We have insufficient artillery power in place and our forces are asking for more support to defend Pisky," he said in a video posted online.
"But the town is basically under Ukrainian control."
In the neighbouring Russian-occupied region of Luhansk, in the grounds below an abandoned, charred apartment block, Lilia Ai-Talatini, aged 48, watched on as her mother's body was exhumed from a makeshift grave to be taken to a cemetery for a proper burial.
Ms Ai-Talatini said how it had taken her 10 days to reach her parents' apartment, which was on the Russian held-side of the town of Rubizhne, during heavy fighting there in March.
"Mother was already dying ... her hands were blue, her complexion was sallow, there were circles under her eyes," she said.
"The next day, mother passed away."
An official with the Luhansk People's Republic, a statelet set up by pro-Russian separatists, said a team had been working in Rubizhne for 10 days and exhumed 104 sets of remains.
"It's clear that shrapnel wounds predominate, but there are also bullet wounds," Anna Soroka said, estimating there were 500 unofficial graves in the city.
These battlefield accounts could not be independently verified.
Russia calls its invasion of Ukraine a "special military operation" to demilitarise and "denazify" its smaller neighbour, while Ukraine and its Western allies regard Moscow's actions as a war of aggression.
The conflict has pushed US-Russian relations to a low point, with Russia warning it may sever ties.
Having been largely isolated on the global diplomatic stage, Russia has been gaining more sympathy from China, whose own ties with Washington have nosedived due to tensions over Taiwan.
And this morning, North Korean state media said Russian President Vladimir Putin told leader Kim Jong Un the two countries would expand "comprehensive and constructive" ties.
Grain ships continue to set sail
Amid the fighting, more ships carrying Ukrainian grain left or prepared to do so as part of a deal struck late last month to ease a global food crisis.
An Ethiopia-bound cargo, the first since Russia's invasion of Ukraine, was getting ready to leave in the next few days, while sources said the first grain ship to leave Ukraine under a UN deal was nearing Syria.
"The world needs the food of Ukraine," Marianne Ward, the deputy country director of the World Food Programme, told reporters.
"This is the beginning of what we hope are normal operations for the hungry people of the world."