Hamas terrorists -- travelling in vehicles, boats and motorised paragliders -- breached Gaza's security barrier and attacked nearby Israeli towns and military posts, opening fire on residents and passersby.
Gun battles raged into the night as at least 70 people were reported killed in Israel, while Gaza authorities released a death count of 232 in the conflict's bloodiest escalation in years which also left many hundreds wounded on both sides.
The army said in the evening its forces were still engaged in live gun battles in 22 locations, in an ongoing operation labelled "Swords of Iron", as reservists were being called up.
"There are still 22 locations where we are engaging with terrorists that came into Israel, from the sea, from the land and from the air," said army spokesman Richard Hecht, adding that the Hamas attack included a "robust ground invasion".
Hamas earlier released images of several Israelis taken captive, and another Israeli army spokesman, Daniel Hagari, confirmed that "there are kidnapped soldiers and civilians.
"I can't give figures about them at the moment. It's a war crime committed by Hamas and they will pay the price."
"We are at war," Israeli Prime Minister Benjamin Netanyahu told the stunned nation as the airforce pounded targets in the blockaded coastal enclave, including several residential tower blocks that were completely destroyed.
"The enemy will pay an unprecedented price," the veteran premier vowed after Hamas had launched its first such combined ground, air and sea offensive, half a century after the outbreak of the 1973 Arab-Israeli war.
The Islamist group started the multi-pronged attack around 6:30 am (0330 GMT) with thousands of rockets aimed as far as Tel Aviv and Jerusalem, some bypassing the Iron Dome defence system and hitting buildings.
"Send help, please!" one Israeli woman sheltering with her two-year-old child pleaded as terrorists outside opened fire and tried to break into their safe room, Israeli media reported.
Bodies were seen lying on the streets of Sderot near Gaza and inside cars, their windscreens shattered by bullets.
"I saw many bodies, of terrorists and civilians," one man who gave his name as Shlomi told AFP, standing beside covered corpses on a road near Gevim Kibbutz in southern Israel.
"So many bodies, so many bodies."
'Gates of hell'
Israeli army Major General Ghasan Alyan warned that Hamas had "opened the gates of hell".
An AFP journalist in Gaza saw smoke billowing from the remains of a residential tower which Gaza's interior ministry said contained 100 apartments and was completely destroyed.
Israel's military said it had warned residents to evacuate before targeting the multi-storey buildings used by Hamas.
The aid group Doctors without Borders said one strike had hit the enclave's Indonesian hospital and an ambulance outside Nasser Hospital in southern Gaza, causing multiple deaths.
As night fell, Israel's state-run electricity company cut the power supply to Gaza.
The escalation follows months of rising violence, mostly in the occupied West Bank, and tensions around Gaza's border and at contested holy sites in Jerusalem.
Hamas labelled its attack "Operation Al-Aqsa Flood" and called on "resistance fighters in the West Bank" as well as in "Arab and Islamic nations" to join the battle.
"We decided to put an end to all the crimes of the occupation (Israel)," said its armed wing, the Ezzedine al-Qassam Brigades, claiming to have fired more than 5,000 rockets.
Hecht said Israel had counted more than 3,000 incoming rockets though the day.
Hamas chief Ismail Haniyeh claimed the group was on the "verge of a great victory and a clear conquest on the Gaza front".
"Enough is enough," he said in a televised address. "The cycle of intifadas (uprisings) and revolutions in the battle to liberate our land and our prisoners languishing in occupation prisons must be completed."
'Dangerous precipice'
Air raid sirens wailed across southern and central Israel, as well as in Jerusalem, where multiple incoming rockets were intercepted by air defence systems.
In Tel Aviv, a gaping hole was ripped into a building, with residents boarding a bus to seek safety in a hotel.
Rocket impacts left cars burning beneath residential buildings in the Israeli city of Ashkelon, north of Gaza.
Among the dead was the president of a regional council for Israeli communities northeast of Gaza, who was killed in a gun battle.
Schools will remain closed on Sunday, the start of the week in Israel.
The conflict sparked major disruption at Tel Aviv airport, with American Airlines, Emirates and Ryanair among carriers with cancelled flights.
Hamas took control of Gaza in 2007, leading to Israel's crippling blockade of the impoverished enclave of 2.3 million people.
Israel and Hamas have since fought several wars. The last major military exchange, in May, killed 34 Palestinians and one Israeli.
In northern Gaza on Saturday, hundreds of people fled their homes, carrying food and blankets, an AFP correspondent said.
Across the West Bank, including annexed east Jerusalem, three Palestinians were killed and around 80 wounded in clashes with Israeli forces and settlers, the Palestinian health ministry and Red Crescent Society said.
In Israeli-annexed east Jerusalem, some Palestinian residents cheered and blew their car horns as sirens blared.
Western capitals condemned the attacks by Hamas, which the United States, European Union and Israel consider a terrorist group.
But Hamas drew support from other foes of Israel, with Iran's supreme leader declaring he was "proud" and Lebanese militant group Hezbollah praising the "heroic operation".
The United States condemned the attacks by "Hamas terrorists" and vowed to ensure the key US ally has the means to defend itself.
European Commission President Ursula von der Leyen called the attack "terrorism in its most despicable form".
UN Middle East peace envoy Tor Wennesland warned of "a dangerous precipice" and called on all sides to "pull back from the brink".
Before Saturday's violence, at least 247 Palestinians, 32 Israelis and two foreigners had been killed this year, including combatants and civilians, according to Israeli and Palestinian officials.
