Violence between Israel and the Palestinians has been surging for almost two years, with fatalities in the occupied West Bank hitting a scale not seen in years.
At least two people were killed in Israel, officials said.
Israel's Defence Minister Yoav Gallant said Palestinian militant group Hamas has "launched a war against the State of Israel."
"Troops are fighting against the enemy at every location," he said in a statement.
AFP journalists said Israel's military began air strikes on Gaza, following the rocket barrage from inside the territory which is sealed off from Israel by a militarised border barrier.
"Dozens of IDF fighter jets are currently striking a number of targets belonging to the Hamas terrorist organisation in the Gaza Strip," the military said.
Rockets had earlier streamed across the sky repeatedly after the first launches from multiple locations across the Palestinian territory from 6:30 am (0330 GMT), AFP journalists in Gaza City reported.
The armed wing of Hamas, which controls Gaza, said it was behind the aerial assault, saying its militants had launched thousands of rockets and its fighters seized an Israeli tank.
Israel's army did not immediately comment on the tank claim when contacted by AFP.
Israeli security chiefs convened over the violence, which occurred on Shabbat and during a Jewish holiday.
Air raid sirens wailed across southern and central Israel, and the army urged people to stay near bomb shelters.
AFP journalists in Jerusalem heard multiple rockets being intercepted by Israeli air defence systems. Sirens blared across the city on more occasions than in any Gaza conflict in the past three years.
"We decided to put an end to all the crimes of the occupation (Israel). Their time for rampaging without being held accountable is over," the Ezzedine al-Qassam Brigades said.
"We announce Operation Al-Aqsa Flood and we fired, in the first strike of 20 minutes, more than 5,000 rockets."
- Hundreds of Gazans flee -
Hundreds of residents fled their homes in eastern Gaza to move away from the border with Israel, an AFP correspondent said.
Men, women and children carrying blankets and food left their homes, mostly in the northeastern part of the territory, the reporter said.
Israel's military said Hamas launched "massive shooting of rockets", while at the same time "terrorists infiltrated into Israeli territory in a number of different locations".
Hamas "will face the consequences and responsibility for these events", it said in a statement.
In Israeli-annexed east Jerusalem, some Palestinian residents cheered and blew their car horns as sirens blared.
A regional council for Israeli communities northeast of Gaza said its president was killed in an exchange of fire with attackers from Gaza.
Separately, a woman in her 60s was killed "due to a direct hit" in Israel, the Magen David Adom emergency services said.
Fifteen others were wounded, two of them seriously, medics said.
An AFP photographer in the coastal city of Tel Aviv saw a gaping hole in a building, with residents gathered outside.
- Hamas calls to 'join battle' -
Hamas called on "the resistance fighters in the West Bank" as well as "our Arab and Islamic nations" to join the battle, in a statement posted on Telegram.
The United States condemned the Hamas fire and urged "all sides to refrain from violence and retaliatory attacks."
"Terror and violence solve nothing," the US Office of Palestinian Affairs wrote on X, formerly Twitter.
Israel has imposed a crippling blockade on Gaza since 2007 after Hamas took power.
Palestinian militants and Israel have fought several devastating wars since.
The latest violence follows heightened tensions in September, when Israel closed the border to Gazan workers for two weeks.
The shutdown of the crossing came as Palestinian demonstrators along the border burned tyres and threw rocks and petrol bombs at Israeli troops, who responded with tear gas and live bullets.
Resuming workers' passage on September 28 had raised hopes of calming the situation in impoverished Gaza, home to 2.3 million people.
In May, an exchange of Israeli air strikes and Gaza rocket fire killed 34 Palestinians and one Israeli.
So far this year at least 247 Palestinians, 32 Israelis and two foreigners have been killed in the conflict, including combatants and civilians on both sides, according to Israeli and Palestinian officials.
The vast majority of fatalities have occurred in the West Bank, which has been occupied by Israel since the 1967 Arab-Israeli War.
There has been a rise in army raids, Palestinian attacks targeting Israelis and Israeli settler violence against Palestinians and their property.
The rising violence this year came against the backdrop of divisive judicial reforms introduced by the hard-right government of President Benjamin Netanyahu, who is on trial for corruption charges he denies.
Several far-right ministers in Netanyahu's cabinet live in West Bank settlements deemed illegal under international law.
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