The city of Kyiv covers 325 square miles (840 square km) and is divided by a broad river. It has about 500,000 structures — factories, ornate churches and high-rise apartments — many on narrow, winding streets. Roughly 2 million people remain after extensive evacuations of women and children.

To the northwest and to the east, tens of thousands of Russian troops are pressing toward the city, Ukraine’s capital, backed by columns of tanks, armored vehicles and artillery. Inside Kyiv, Ukrainian soldiers and civilian volunteers are fortifying the downtown with barriers, anti-tank mines and artillery.

Kyiv remains the biggest prize of all for the Russian military; it is the seat of government and ingrained in both Russian and Ukrainian identity. But capturing it, military analysts say, would require a furious and bloody conflict that could be the world’s biggest urban battle in 80 years.