Iran said "the gates of hell will open" in retaliation while Israel said the strikes were only the start of its campaign.
US President Donald Trump said it was not too late for Iran to halt the Israeli attacks by reaching a deal to curb its nuclear program.
As evening fell on Friday, Iranian media reported a number of explosions in what appeared to be a second wave of strikes.
"Moments ago, Israel launched Operation Rising Lion, a targeted military operation to roll back the Iranian threat to Israel's very survival," Prime Minister Benjamin Netanyahu said early on Friday in a televised address that invoked the failure of the world to prevent the Holocaust in World War II.
Israel's operation "will continue for as many days as it takes to remove this threat," he said.
"Generations from now, history will record our generation stood its ground, acted in time and secured our common future."
Iran's Supreme Leader Ayatollah Ali Khamenei said Israel had "unleashed its wicked and bloody" hand, and would suffer "a bitter fate".
In a phone interview with Reuters, Trump said it was not clear if Iran's nuclear program had survived but it was not too late for Iran to make a deal to halt the Israeli assault.
He said nuclear talks between Iran and the United States, scheduled for Sunday, were still on the agenda although he was not sure if they would take place.
"I tried to save Iran humiliation and death," Trump said.
Earlier, he posted on Truth Social: "Iran must make a deal, before there is nothing left."
Two regional sources said at least 20 Iranian military commanders were killed, a stunning decapitation reminiscent of Israeli attacks that swiftly wiped out the leadership of Lebanon's once-feared Hezbollah militia last year.
Among the generals killed on Friday were the armed forces chief of staff, Major General Mohammad Bagheri and the Revolutionary Guards chief Hossein Salami.
Major General Mohammad Pakpour, swiftly promoted to replace Salami as Guards commander, vowed retaliation in a letter to the Supreme Leader read out on state television: "The gates of hell will open to the child-killing regime."
Twenty people Reuters spoke to inside Iran described an atmosphere of fear and anger, with some people rushing to change money and others seeking a way out of the country to safety.
"People on my street rushed out of their homes in panic, we were all terrified," said Marziyeh, 39, who was awakened by a huge blast in Natanz, a city that houses the huge underground facility where Iran purifies the uranium that Israel said could be used to make a nuclear bomb.
Iranian media showed images of destroyed apartment blocks, and said nearly 80 civilians were killed in attacks that targeted nuclear scientists in their beds and wounded more than 300 people.
Iran's main nuclear enrichment facility at Natanz suffered significant damage, Israeli military spokesman Brigadier General Effie Defrin said.
Briefing journalists online, Defrin said 200 Israeli fighter jets took part in the strikes, hitting more than 100 targets, with others to work through, and that the operation might be lengthy.
The United Nations Security Council was due to meet on Friday at Iran's request.
Iran said in a letter to the Council that it would respond decisively and proportionally to Israel's "unlawful" and "cowardly" acts.
An Israeli security source said Mossad commandos had been operating deep inside Iran before the attack, and the Israeli spy agency and military had mounted a series of covert operations against Iran's strategic missile array.
Israel also established an attack-drone base near Tehran, the source added.
The military said it had bombarded Iran's air defences, destroying "dozens of radars and surface-to-air missile launchers".
Israeli officials said it may be some time before the extent is clear of damage to the underground nuclear site at Natanz, where Iran has refined uranium to levels some countries have long said are suitable for a bomb rather than civilian use.
Iran has long insisted its nuclear program is for civilian purposes only.
Israel remained on alert, shutting its embassies around the world and urging citizens via embassy websites to stay vigilant and avoid displaying Jewish or Israeli symbols in public.
Israeli military Chief of Staff Eyal Zamir said tens of thousands of soldiers had been called up and "prepared across all borders".
Oman, which is mediating in Iran-US nuclear talks, described Israel's attack as reckless and urged the international community to halt the "dangerous course of action".
