The ceasefire, agreed on Friday, held for much of Saturday, but shelling started overnight.
The official Twitter account of the Donetsk rebels said in the early hours of Sunday that its forces were "taking Mariupol", but later accused Ukraine of breaking the ceasefire.
Fighters from the Azov battalion, who are defending the town, said their positions had come under Grad rocket fire.
Earlier on Saturday the truce had appeared to be holding, with only minor violations reported, as hopes mounted that the deal struck in Minsk on Friday could bring an end to the violence that has left more than 2,000 dead in recent months.
Both sides accused the other of violating the ceasefire, but there did not appear to be any serious exchanges of fire and no casualties were reported.
Nevertheless, the rhetoric coming from Kiev and Donetsk, capital of the Russia-backed rebel movement, showed that a political solution was still some way away.
The atmosphere between the two frontlines on Saturday was tense but calm, as both sides took stock of what appear to have been heavy losses in the final fighting that led up to the ceasefire.
The fiercest fighting on Friday came in the villages between Novoazovsk and Mariupol, the strategic port city that Ukrainians feared would be attacked by separatists over the past week.
Rebel forces seized the town of Novoazovsk, across the border with Russia, 10 days ago.
Kiev says the rebels were aided by soldiers and armour of the regular Russian army, which helped turn the tide against Ukraine's forces and push Kiev towards accepting a ceasefire.
At one point on the main road, a Ukrainian tank had been hit so hard on Friday it had been thrown back on to a huge stone wave-breaker placed on the road, its treads spooling out behind it.
Fields all around were scorched, and in some places smoke was still rising from where Grad missiles had landed the day before.
In a nearby village, three Ukrainian tanks had been abandoned in the courtyard of a school kindergarten.
Two were burned out, while one was untouched but had clearly been left in a hurry, rucksacks and personal possessions of its occupants left strewn around it, a sticker reading "Fuck off Putin" stuck to the base of its turret.
The windows of the kindergarten had all been blown out, its roof removed, and there were gaping holes in the walls, apparently from mortar rounds.
"The tanks came in about six in the morning," said one villager who did not want to give his name.
"As soon as they set up position there, you could hear the booms come in from that direction," he said, pointing at the pro-Russia lines.
It was unclear whether those in the tanks had escaped alive.
There was anger in the village, which until Saturday had seen no major fighting, over the destruction of the school.
"See what a glorious army we have," said one middle-aged woman, sarcastically.
"Parked their tanks up in our kindergarten, now the whole thing has gone."
In Mariupol, there was a more-relaxed atmosphere, as for the first day in a week there was no suggestion that rebel forces might attempt to enter.
The residents are far from unanimous supporters of the Kiev government; many here would have preferred the region to be annexed by Russia.
But most simply want peace.
Fighters from the Azov battalion, the volunteer grouping with far-right leanings that has done much of the fighting around Mariupol, sat on a restaurant terrace eating pizzas; families strolled in the sunshine, wedding parties breezed through central streets beeping horns.
Ukraine's President Petro Poroshenko spoke by telephone with Vladimir Putin on Saturday on ways to make the ceasefire last, but the final political solution remains unclear.
Alexander Zakharchenko, prime minister of the so-called Donetsk People's Republic, travelled to Moscow from the negotiations, and told journalists on Saturday that the peace talks had "legitimised" his quasi-country, and it should be regarded as on an equal footing with Ukraine.
Analysts have suggested that Moscow is likely to push either for a "breakaway state" which will be unrecognised internationally but function independently, or for east Ukraine to be de facto part of Ukraine but with such autonomy that it can essentially become a proxy region for Russia, ensuring that the rest of Ukraine can never join NATO or fully orient its foreign policy westward.
On Saturday, Patriarch Filaret, the head of the Kiev Patriarchate, accused Putin of being under Satan's influence.
"This ruler is cynically lying, saying his country is not a party to the conflict in Ukraine, though he did everything in order to foment the conflict and maintain it," said Filaret in a statement.
"He calls himself a brother to the Ukrainian people, but in fact according to his deeds, he has really become the new Cain, shedding the brotherly blood and entangling the whole world with lies."
Source: The Guardian Today