Ukraine's top commander has said Kyiv's forces control 1,000 sq km of Russian territory as they press their biggest cross-border incursion in two-and-a-half years of full-scale war.
Commander Oleksandr Syrskyi said Ukraine continued to "conduct an offensive operation in the Kursk region" seven days after it began.
Ukraine's President Volodymyr Zelensky said Russia had brought war to others and now it was coming back to Russia.
But Russian leader Vladimir Putin described the offensive as a "major provocation" and ordered Russian forces to "to kick the enemy out of our territory".
A growing number of people have been evacuated from the western Russian region for their safety, with a further 59,000 told to leave.
The local governor said some 28 villages in the area had fallen to Ukrainian forces, that 12 civilians had been killed and that "the situation remains difficult".
Ukrainian troops launched their surprise attack last Tuesday, advancing up to 18 miles (30km) into Russia.
The offensive is said to have boosted morale on the Ukrainian side, but analysts say the strategy brings fresh dangers to Ukraine.
A senior British military source, who asked not to be named, told the OceanNewsUK there was the risk that Moscow will be so angered by this incursion that it could redouble its own attacks on Ukraine’s civilian population and infrastructure.
In comments aired on state television, President Putin said on Monday: "One of the obvious goals of the enemy is to sow discord, strife, intimidate people, destroy the unity and cohesion of Russian society.
"The main task is, of course, for the defence ministry to dislodge the enemy from our territories," he told a meeting of officials.
The region's governor said 121,000 people had been evacuated from their homes. He told Mr Putin that about 2,000 Russian citizens remained in areas occupied by Ukrainian forces in the area.
"We don't know anything about their fate," he said.
He warned people to take shelter from missiles in rooms without windows and with solid walls.
In Belgorod, the region next to Kursk, about 11,000 people were also urged to leave, as governor Vyacheslav Gladkov told people from the Krasnaya Yaruga district they were being moved due to "enemy activity on the border".
He issued a similar missile warning, and told people to shelter in their basements.
In his nightly address, Ukraine's president acknowledged the offensive, saying: "Russia must be forced to make peace if Putin wants to fight so badly."
"Russia brought war to others, now it's coming home. Ukraine has always wanted only peace, and we will certainly ensure peace," Mr Zelensky added.
Ukrainian officials have said thousands of troops are engaged in the operation, far more than the small incursion initially reported by Russian border guards.
An official told news agency AFP that their aim was to "to inflict maximum losses and to destabilise the situation in Russia".
During a meeting with Mr Zelensky in Kyiv on Monday, US Senator Lindsey Graham called the cross-border operation "brilliant" and "bold", and urged the Biden administration to provide Ukraine with the weapons it needs.
Some in Russia have questioned how Ukraine was able to enter the Kursk region - with one pro-Russian war blogger, Yuri Podolyaka calling the situation "alarming".
Foreign Ministry spokeswoman Maria Zakharova said a tough response from Russia's armed forces "will not take long".
Meanwhile, Russian ally Belarus said it was bolstering its own troop numbers at its border after claiming Ukraine had entered its airspace with drones.
