The camps had been occupied by supporters of former president Mohammed Morsi, who was deposed in early July.
The health ministry says 149 people have been killed. But the Muslim Brotherhood, which backed the protests, says at least 2,000 have died.
The state of emergency is scheduled to last for a month.
A curfew will be in place in 11 provinces, including Cairo, starting at 19:00 (17:00 GMT).
The measure was taken because the “security and order of the nation face danger due to deliberate sabotage, and attacks on public and private buildings and the loss of life by extremist groups,” the presidency said in a statement.
Shortly after dawn on Wednesday morning, armoured bulldozers moved deep into the main protest camp outside the eastern Rabaa al-Adawiya mosque.
Officials say the other protest camp, at Nahda Square, has been cleared.
Reporters described wounded protesters being treated next to the dead in makeshift field hospitals.
The 17-year-old daughter of leading Muslim Brotherhood figure Mohamed el-Beltagy was among the dead, reports say. Asmaa el-Beltagy was shot in the back and chest, her brother said.
A cameraman working for Sky News, Mick Deane, has also been killed – as has a reporter for Gulf News, Habiba Ahmed Abd Elaziz.
The White House condemned the bloodshed, saying the violence “runs directly counter to the pledges by the interim government to pursue reconciliation”.
UN Secretary General Ban Ki-moon urged “all Egyptians to concentrate their efforts on promoting genuinely inclusive reconciliation”, his spokesman Martin Nesirky said.
There are also reports of unrest elsewhere in Egypt.
$1· About 35 people have died in clashes in the province of Fayoum, south of Cairo, Reuters news agency says.
$1· At least five people have been killed in the province of Suez, according to the health ministry.
$1· Clashes have also been reported in the northern provinces of Alexandria and Beheira, and the central provinces of Assiut and Menya
$1· Hundreds are said to have gathered outside the governor’s office in Aswan in the south
$1· State news agency Mena says three churches were attacked, one in the city of Sohag with a large number of Coptic Christian residents
The interior ministry said a mopping-up operation in the streets surrounding Nahda Square was under way.
Pro-Morsi activists were chased into the nearby zoo and Cairo University, Nile TV said.
It is still unclear how many casualties were caught up in the two Cairo operations. Figures differ widely and have been impossible to verify independently.
BBC Arabic’s Khaled Ezzelarab says he counted at least 50 bodies at the makeshift hospitals around Rabaa al-Adawiya mosque. He said the injured were too numerous to count.
Ikhwanonline, the website of the Muslim Brotherhood, says that in total more than 800 were killed.
The health ministry has issued an official death toll of 95.
The interior ministry denied any deaths were caused by its forces firing live ammunition.
“Security forces used only tear gas canisters to disperse the protesters though it was heavily fired at by armed elements from inside the two protest camps, causing the death of an officer and a conscript and the injury of four policemen and two conscripts,” the ministry said in a statement.
The government has meanwhile congratulated the security forces on their operation to clear the camps.
In a televised statement, a government spokesman praised their “self-restraint” and spoke of the “smaller number” of injuries among protesters.
The government would decisively confront attempts to attack state buildings and police stations, he said.
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Supporters of Mr Morsi have been occupying Nahda Square and the Rabaa al-Adawiya site since he was ousted on 3 July. They want him reinstated.
Large plumes of smoke rose over parts of the city as the operation to clear the camps began, with tear gas canisters fired and helicopters hovering above.
Muslim Brotherhood TV called for people to send cars to the sit-ins to take casualties to hospital.
Several Muslim Brotherhood figures have been arrested, security sources said.
The protesters had been expecting the clearance operation, says BBC Middle East editor Jeremy Bowen.
It is a heavy-handed operation and is a consequence of the current “winner takes all” climate, he adds, with both sides sticking to their positions and pushing as hard as they can.
Call for restraint
There has been strong international reaction to the storming of the camps.
White House spokesman Josh Earnest told a news conference that the administration had repeatedly urged the Egyptian military and security forces to show restraint, and strongly opposed the declaration of a state of emergency.
The European Union called the reports of deaths and injuries “extremely worrying”.
A statement issued on behalf of EU foreign policy chief Catherine Ashton said: “We reiterate that violence won’t lead to any solution and we urge the Egyptian authorities to proceed with utmost restraint.”
Turkish PM Recep Tayyip Erdogan described the result of the camp clearances as a massacre, accused other countries of paving the way for the violence by staying silent, and called for the UN and the Arab League to act immediately.
UK Foreign Secretary William Hague also condemned the use of force.