There has been fresh shelling near Donetsk airport in eastern Ukraine, raising fears that a recently agreed ceasefire may collapse.
The truce held for much of Saturday but shelling in Mariupol, which killed one woman, was followed by the Donetsk airport blasts early on Sunday.
The two cities then turned quiet, with no reports of clashes overnight.
Fighting in the east has killed some 2,600 people since April. The truce and roadmap to peace were agreed on Friday.
On Sunday, Ukrainian security official Volodymyr Poliovyi said 864 Ukrainian soldiers had been killed since the conflict began.
Russia has repeatedly denied accusations by Ukraine and the West that it has been sending troops into Ukraine's eastern Donetsk and Luhansk regions to help the rebels, who want to establish an independent state.
Before the ceasefire was agreed in Minsk, Belarus, on Friday, the separatists had been advancing on both Donetsk airport and Mariupol, a key city on the route to Crimea, which was annexed by Russia in March.
At the scene: BBC's Richard Galpin in Donetsk
We have just been at Donetsk airport which has in recent weeks been the focus of the fighting between Ukrainian forces and separatist rebels, with Ukrainian troops inside the main building there holding the airport.
We have heard a lot of mortars being fired, both into the airport area and outgoing as well. We also met an observer team from the Organization for Security and Co-operation in Europe (OSCE).
It said most of the firing was outgoing from the airport, ie from Ukrainian troops. It said the firing had caused some fires in villages nearby but stressed that, although a violation of the ceasefire, the fighting at Donetsk airport was not sufficient to actually make the ceasefire collapse.
The presidents of Ukraine and Russia both said on Saturday that the truce was largely holding.
However, late on Saturday, eyewitnesses said pro-Russian rebels had shelled and destroyed a government checkpoint on the eastern approaches to Mariupol - a major port on the Azov Sea.
Local officials said one 33-year-old woman civilian was killed and three people injured in the shelling.
One Ukrainian soldier told Reuters that government forces had pulled out tanks in line with the truce.
He said: "We only left lightly armed people to man checkpoints and these monsters violated every word of the agreement."
However, separatist leader Andrei Purgin told Russia's RIA news agency: "Despite the provocations of Ukrainian forces, the militias... will keep firmly to the Minsk agreement. The militias are not resorting and will not resort to arms."
One Mariupol resident told Agence France-Presse: "I'm frightened. I want peace but I think this ceasefire is finished; this is the third night we haven't been able to sleep."
The BBC's Fergal Keane in Mariupol reported renewed shelling there on Sunday. He says it lasted only a short time but the situation on the city's eastern fringes remains tense, and there is little confidence the conflict has ended.
The BBC's Richard Galpin reported fresh fighting on visiting Donetsk airport. It has been under the control of government troops since May but has been under constant rebel attack.
One rebel fighter told Reuters news agency: "Listen to the sound of the ceasefire. There's a proper battle going on there."
Meanwhile, Yuri Lutsenko, who advises Ukrainian President Petro Poroshenko, said a deal was struck at the Nato summit in Wales to supply Ukraine with "modern weapons from the US, France, Italy, Poland, Norway".
A US official was later quoted by Reuters as saying that "no US offer of lethal assistance has been made to Ukraine".
Defence officials in Italy, Poland and Norway also reportedly denied plans to provide weapons.
Separately, a new report by Amnesty International accuses all sides in the conflict of committing war crimes.
The human rights group said civilians had accused Ukrainian government troops of shelling their neighbourhoods indiscriminately.
Witnesses also said that separatist fighters had "abducted, tortured, and killed their neighbours". The report accused Russia of fuelling separatist crimes.
12-point peace roadmap - key elements
- Ensure an immediate bilateral ceasefire
- Carry out decentralisation of power, allowing temporary local self-government in areas of Donetsk and Luhansk in eastern Ukraine under a "special status" law
- Immediately free all hostages and illegally detained persons
- Ensure monitoring on the Ukrainian-Russian border and a security zone
- Ensure the holding of snap local elections in Donetsk and Luhansk
- Remove illegal armed groups, military hardware, and all fighters and mercenaries from Ukrainian territory
- Pass a law against the prosecution and punishment of people over certain events in Donetsk and Luhansk region
Posted by the OSCE on its website (in Russian).