When the strikes began on Saturday morning, explosions rocked Tehran and columns of smoke rose into the sky, shaking the city at the start of the Iranian working week.
Israel said Supreme Leader Ayatollah Ali Khamenei had been killed in the operation, but Iranian authorities have not confirmed his fate.
Forty people were killed in an Israeli air strike on a school, state media said. Reuters could not independently confirm the reports.
Residents reached by phone described scenes of chaos and alarm as they rushed to collect their children from school or made preparations to leave home.
"We are going to our hometown in Yazd, Tehran is not safe anymore. They said roads are safe, but I am worried," said Gholamreza, a Tehran shopkeeper and father of two.
"I am leaving everything behind in Tehran," said Gholamreza.
It marks the latest upheaval for Iranians, weeks after thousands of people were killed in a government crackdown on nationwide unrest, and comes just eight months after last year's 12-day war with Israel, during which the US bombed Iranian nuclear sites.
Iran's top security body said it expected attacks to continue on Tehran and some other cities, and urged people to "travel to other cities where possible so that you may remain safe from the harm of these two regimes' acts of aggression".
Schools and universities are closed until further notice.
"We are scared, we are terrified. My children are shaking, we have nowhere to go, we will die here," said Minou, a 32-year-old mother of two from the northern city of Tabriz, one of many areas where explosions were reported.
US President Donald Trump said the operation would end a security threat to the United States and offer Iranians a chance to topple their rulers.
An Iranian from the central city of Yazd said he hoped the attack would topple the clerical establishment that has run the country since the 1979 Islamic Revolution. "Let them bomb," the resident of Yazd said.
Samira Mohebbi, speaking from the northern city of Rasht, disagreed.
"I am against this regime, to hell with them. But I don't want my country to be attacked by foreign forces, I don't want my Iran to turn into Iraq," she said, referring to the neighbouring country that suffered years of chaos and bloodshed following the US-led invasion that toppled Saddam Hussein.
'Security forces blocked roads in the area of Tehran that is home to the offices of Khamenei, President Masoud Pezeshkian and parliament, witnesses said.
In Tabriz, two residents said people were rushing to supermarkets to buy food, fuel and generators.
"They said the nuclear talks are going well. They fooled us again," said a resident of Tehran.
Western governments have long suspected Iran aims to build a nuclear bomb. Tehran has always denied this.
