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2 year oldKherson is the first major city to be taken by Russia, after heavy fighting, since it invaded a week ago.
Its mayor, Igor Kolykhaev, said Russian troops had forced their way into the city council building and imposed a curfew on residents.
Several cities have come under intense shelling, with Wednesday one of the most destructive days of the fighting.
An investigation into possible war crimes and crimes against humanity committed in Ukraine has been launched by the International Criminal Court (ICC) in The Hague.
Russia has for the first time admitted taking heavy military casualties during its attack on Ukraine, with 498 troops killed and a further 1,597 injured. Ukraine says Russia's losses run into the thousands.
Ukraine reports that more than 2,000 civilians have died since the invasion began last Thursday. The conflict has also caused more than a million people to flee Ukraine, according to the UN.
In a Facebook post, Mr Kolykhaev said Russian forces were in control of Kherson, a port on Ukraine's southern Black Sea coast with a population of more than 280,000 people.
He urged Russian soldiers not to shoot at civilians, saying there were no Ukrainian forces in the city.
Mr Kolykhaev called on residents to follow conditions set by Russian forces in order to "keep the Ukrainian flag flying". These include:
"The (Russian) occupiers are in all parts of the city and are very dangerous," Gennady Lakhuta, head of the regional administration, was quoted as saying by AFP news agency.
The capture of Kherson - located on the banks of the Dnieper River where it flows into the Black Sea - is significant because it could allow Russia to create a base for the military there as it seeks to push further inland.
Ukraine's second-largest city, Kharkiv, came under a fierce aerial assault. Its mayor told the BBC shelling and cruise missile strikes were hitting residential areas and inflicting heavy civilian casualties.
And in the southern port of Mariupol, hundreds are feared dead following hours of sustained shelling.
But Russian efforts to encircle the capital Kyiv have slowed, with one US official saying a huge Russian convoy to the north of the city barely moved all day on Wednesday.
Large explosions were heard in the capital overnight into Thursday, with footage showing a huge fireball lighting up the night sky.
Biggest blasts we've ever seen, just as we were coming off air tonight in #Kyiv #cbsnews @charliecbs @cbsnews pic.twitter.com/leIfTrdnXx
— Justine Redman (@redmanjustine) March 3, 2022
Four blasts were heard in the city centre, near Friendship of the Peoples metro station, Ukrainian media report.
In other developments:
Pictures of fighting in Ukraine have prompted international condemnation of Russia, with an overwhelming majority of UN countries calling on Moscow to immediately withdraw its troops.
Only five countries voted against the resolution, including Syria and North Korea.
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