Ukraine pressed ahead with its assault inside Russian territory on Tuesday, a week into the biggest foreign incursion into the country since World War II.

The cross-border attack caught Russia by surprise and signified a shift in tactics for Kyiv, more than two years after Moscow’s troops poured across Ukraine’s border in a full-scale invasion.

The rapid advance by Ukrainian forces has been an embarrassment for the Kremlin and aims to alter the narrative of the war at a time when Kyiv’s forces are stretched thin on the front lines of their own country.

Here’s what to know about Ukraine’s cross-border operation, which President Biden said was creating a “real dilemma” for the Russian government.

What happened?

Ukrainian troops and armored vehicles stormed into the Kursk region of western Russia the morning of Aug. 6, punching through border defenses and seizing several settlements in heavy fighting that left a trail of death and ruin.

The assault opened a new front in the 30-month war and did not just catch Russia off guard: Some Ukrainian soldiers and U.S. officials also said they lacked advance notice.

Russia’s top general estimated that Ukraine had deployed 1,000 soldiers for the incursion, while U.S. officials said Ukraine had sent several thousand. Military analysts have said the attack involved elements of at least four brigades in a rare example of a successful operation involving support from artillery, air defenses and electronic warfare. That translated into quick advances on the ground.

How far into Russia have Ukrainian troops advanced?

Ukrainian forces advanced several miles into Russia within the first 24 hours of the incursion.

By Monday, Aleksei Smirnov, the Kursk region’s acting governor, said that 28 towns and villages were under Ukrainian control. Ukrainian troops had pushed seven miles into Russian territory along a 25-mile front, he said, adding that 12 civilians had died in the fighting.

His claims could not be independently verified, though the description of the extent of Ukraine’s advance was roughly in line with analysts’ estimates.

The head of Ukraine’s armed forces, Gen. Oleksandr Syrsky, on Monday claimed Kyiv had control of “about 1,000 square kilometers,” or just under 400 square miles.

On Tuesday, President Volodymyr Zelensky of Ukraine wrote on X that “our forces continue to advance in the Kursk region.”

Why is this significant?

Kyiv has regularly bombarded Russian oil refineries and airfields with a fleet of homemade drones since Moscow’s full-scale invasion began. It has also helped stage two other ground attacks. Those, however, were smaller forays by Russian exile groups backed by the Ukrainian Army, and they ended in quick retreats.

Until last week, Ukrainian forces had not counterattacked into Russia. The gains in Kursk are the quickest for Ukrainian forces since they reclaimed the Kherson region of their own country in November 2022.

How has the Kremlin responded?

As Ukrainian forces pushed deeper into Russia, Moscow scrambled to shore up its defenses and President Vladimir V. Putin convened his security services to coordinate a response.

The Russian military said it was sending more troops and armored vehicles to try to repel the attack, with Russian television broadcasting images of columns of military trucks.

While the efforts appear to have helped stall further advances by Ukrainian troops, Kyiv’s forces are holding ground a week into the incursion.

Russian officials and the state news media have repeatedly claimed to have the situation under control — most recently on Tuesday — only to then lose more territory.

And what about Putin?

The incursion has embarrassed Mr. Putin and his military establishment, prompting questions about Russia’s level of preparedness.

Underscoring how the attack rattled the Kremlin, Mr. Putin lashed out at the West in a tense televised meeting with his top officials on Monday.

“The West is fighting us with the hands of the Ukrainians,” he said, repeating his frequent depiction of the war, which he started, as a proxy campaign against Russia by the West.

Mr. Putin directed his military to push out Ukraine’s troops and to work with the border guard service to “ensure the reliable protection of the state border” — an acknowledgment that Russia had failed in that regard.

What has Ukraine said?

It took days for Ukraine to publicly acknowledge the cross-border operation, with the military staying silent in the face of accusations and statements from Russian officials.

Mr. Zelensky acknowledged the operation publicly for the first time on Saturday, without explicitly mentioning Kursk — he only did so two days later, saying Russia had launched attacks from the area.

He also hinted at another rationale, adding: “Russia brought war to others, and now it’s coming home.”

What is the goal of Ukraine’s incursion?

Analysts say that Ukraine’s offensive has two main aims: to draw Russian forces from the front lines in eastern Ukraine and seize territory that could serve as a bargaining chip in future peace talks.

The operation could also offer a much-needed morale boost for Ukrainians, whose forces have been losing ground to Russian troops for months.

But military analysts have questioned whether Kyiv’s cross-border assault is worth the risk, given that Ukrainian forces are already stretched on the front lines of their own country.

How is it affecting the fight inside Ukraine?

Russian forces have been pummeling Ukrainian forces in the east even as Moscow races to respond to the incursion into the Kursk region, according to Ukrainian military officials.

Russia has begun to withdraw small numbers of troops from Ukraine to try to help repel the Ukrainian incursion, a U.S. official said on Tuesday without describing how many Russian troops were involved or what impact the reinforcements might have. The official spoke on the condition of anonymity to discuss sensitive security matters.

While Kyiv’s allies have in the past been wary that Ukrainian incursions in Russia could escalate the war, the European Union’s top diplomat, Josep Borrell Fontelles, said that Ukraine had the bloc’s “full support.”

The United States has suggested that the assault in the Kursk region does not violate American guidance, and Mr. Biden told reporters on Tuesday that his administration had been in “constant contact” with Ukraine.

However, senior American officials have said privately that they understood Kyiv’s need to change the narrative of the war, but that they were skeptical Ukraine could hold the territory long enough to force Russia to divert significant forces from the front lines in eastern and southern Ukraine.

What happens next?

Heorhii Tykhyi, a spokesman for Ukraine’s Foreign Ministry, told local news outlets on Tuesday that the country was “not interested in taking territory in Kursk region,” echoing Mr. Zelensky in saying the incursion was aimed at protecting its own citizens.

But it still remains unclear whether Ukraine will try to solidify control over the land it has captured or be forced to retreat.

Russian officials have warned that the incursion could expand. Mr. Smirnov warned on Tuesday that “the crisis has not yet been overcome,” while Vyacheslav Gladkov, the Belgorod region’s governor, said early Wednesday that he was declaring a state of emergency.