Hamas said all of its security compounds in Gaza were destroyed by the air strikes, which Israel said hit some 40 targets.
Hamas vowed to carry out revenge attacks on Israel and fired Qassam rockets into Israeli territory as an immediate reply.
One Israeli was killed by a rocket strike on the town of Netivot, 20 kilometres (12 miles) east of Gaza, doctors said.
The air strikes come amid rumours that an Israeli ground operation is imminent.
Israeli television said on Saturday evening that Israeli troops were massing on the Gaza border "in preparation for a supplementary ground offensive". The report has not been confirmed by independent sources.
US Secretary of State Condoleezza Rice accused Hamas of having triggered the new bout of violence.
Israeli Prime Minister Ehud Olmert: 'We are not fighting against the people of Gaza'
"The United States is deeply concerned about the escalating violence in Gaza," she said in a statement.
"We strongly condemn the repeated rocket and mortar attacks against Israel and hold Hamas responsible for breaking the ceasefire and for the renewal of violence there. The ceasefire must be restored immediately and fully respected."
UN Secretary-General Ban Ki-moon also urged an immediate halt to the violence, condemning what he called Israel's "excessive use of force leading to the killing and injuring of civilians" and "the ongoing rocket attacks by Palestinian militants".
Calls for a ceasefire also came from Middle East envoy Tony Blair and the French EU presidency.
Hamas bases destroyed
Palestinian militants frequently fire rockets against Israeli towns from inside the Gaza Strip; large numbers of rocket and mortar shells have been fired at Israel in recent days.
A resident describes the attacks in Gaza
A Hamas police spokesman, Islam Shahwan, said one of the Israeli raids targeted a police compound in Gaza City where a graduation ceremony for new personnel was taking place.
At least a dozen bodies of men in black uniforms were photographed at the Hamas police headquarters in Gaza City.
Most of the dead and injured were said to be in Gaza City. The head of Gaza's police forces, Tawfik Jaber, was among those killed.
Mr Olmert appealed to Palestinians in Gaza, saying "You - the citizens of Gaza - are not our enemies. Hamas, Jihad and the other terrorist organisations are your enemies, as they are our enemies.
"They have brought disaster on you and they try to bring disaster to the people of Israel. And it is our common goal to make every possible effort to stop them."
It is the worst attack in Gaza since 1967 in terms of the number of Palestinian casualties, a senior analyst told the BBC in Jerusalem.
Mosques issued urgent appeals for people to donate blood and Hamas sources told the BBC's Rushdi Abou Alouf in Gaza that hospitals were soon full.
In the West Bank, Palestinian Authority President Mahmoud Abbas - whose Fatah faction was ousted from Gaza by Hamas in 2007 - condemned the attacks and called for restraint.
Egypt opened its border crossing to the Gaza Strip at Rafah to absorb and treat some of those injured in the south of the territory.
Palestinians staged demonstrations in the West Bank cities of Ramallah and Hebron, and there were some scuffles with Israeli troops there.
Although a six-month truce between Hamas and Israel was agreed earlier this year, it was regularly under strain and was allowed to lapse when it expired this month.
Hamas blamed Israel for the end of the ceasefire, saying it had not respected its terms, including the lifting of the blockade under which little more than humanitarian aid has been allowed into Gaza.
Israel said it initially began a staged easing of the blockade, but this was halted when Hamas failed to fulfil what Israel says were agreed conditions, including ending all rocket fire and halting weapons smuggling.
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