Explosions also rang out in nearby oil-producing Gulf Arab countries, which said they had intercepted missiles from Iran.
The first wave of strikes in what the Pentagon named "OPERATION EPIC FURY" mainly targeted Iranian officials, a source familiar with the matter said.
An Israeli official said Iranian Supreme Leader Ayatollah Ali Khamenei and President Masoud Pezeshkian were both targeted, but the result of the strikes was not clear.
An Iranian source close to the establishment said several senior commanders in Iran's Revolutionary Guards and political officials had been killed.
Forty people were killed in an Israeli air strike on a school, state media said. Reuters could not independently confirm the reports.
The renewed confrontation between Iran and its long-time foes dimmed hopes of a diplomatic solution to Tehran's nuclear dispute with the west.
The latest indirect talks between the US and Iran this week failed to produce a breakthrough.
Iran's Revolutionary Guards said all US bases and interests in the region were within Iran's reach and that the retaliation would continue until "the enemy is decisively defeated".
Iran's foreign minister told counterparts from Saudi Arabia, the UAE, Qatar, Kuwait, Bahrain and Iraq that Tehran would use all its defensive and military capabilities to defend itself.
Loud booms sounded in the United Arab Emirates capital, Abu Dhabi, an oil producer and US ally.
Fighter jets flew around the Yas Island area of Abu Dhabi on Saturday afternoon and blasts were heard in the country's business capital, Dubai.
Bahrain said the service centre of the US Fifth Fleet had been subjected to a missile attack.
Video footage from witnesses in Bahrain showed a thick grey plume of smoke rising from near the small island state's coastline as sirens wailed.
Fellow Gulf Arab state Qatar said it had downed all missiles targeting the country and that it had a right to respond. Sirens were later heard in its capital Doha.
Explosions were heard near Iran's Kharg Island. Iran exports 90 per cent of its crude oil via Kharg, for shipping through the narrow Strait of Hormuz.
Global airlines cancelled flights across the Middle East and the attacks raised the prospect of oil prices rising.
In a video message published on social media, Trump cited Washington's decades-old dispute with Iran, including the seizure of the 1979 US embassy in Tehran, when students held 52 Americans hostage for 444 days, as well as a range of other attacks the US has blamed on Iran since the 1979 Islamic revolution brought the clerics to power.
He urged Iranians to stay sheltered because "bombs will be dropping everywhere".
But he also added: "When we are finished, take over your government. It will be yours to take. This will be probably your only chance for generations."
Israeli Prime Minister Benjamin Netanyahu said the joint US-Israeli attack "will create the conditions for the brave Iranian people to take their destiny into their own hands" and " remove the yoke of tyranny".
In Tehran, witnesses said people were rushing to banks to withdraw cash.
Long queues formed at gas stations across cities.
Many also worried about a potential internet blackout that would cut off communication with their families abroad.
"We are being killed by the regime and by Israel. We are the victims of this regime's hostile policies," said Maryam, 54, a housewife in Tehran, as she headed to northern Iran with her family.
