Kherson 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 1,000,000 people to flee Ukraine, the UN's High Commissioner for Refugees Filippo Grandi said.
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:
- Observing a strict curfew from 20:00 to 06:00
- Only going out in groups of two at most
- Allowing only cars carrying food, medicine and other supplies to enter the city, driving at minimum speed
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.