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Read more: How Hamas staged lightning assault no one thought possible. For the basics: What is Hamas?
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Read more: How Hamas staged lightning assault no one thought possible. For the basics: What is Hamas?
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Edited by Tiffany Wertheimer and James FitzGerald
All times stated are UK
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Hamas claims capture of Israeli soldiers
Graeme Baker
Live reporter
More now on reported hostage takings.
Hamas' military wing, the Qassim Brigades, has released a video showing at least three men in civilian clothes being held captive by militants, claiming the trio are Israeli soldiers.
The video, seen on social media, claims the captives to be "enemies" captured during the offensive.
The AFP news agency says the video is preceded by text, in Arabic, stating "Images of al-Qassim Brigades capturing several enemy soldiers during" the Palestinian operation.
Signs in Hebrew in the background suggest that the video was filmed on the Israeli side of the Erez crossing between Israel and the Gaza Strip, AFP says.
BBC News has so far been unable to independently verify the video.
Complicated situation surrounding reported kidnappings
Frank Gardner
Security correspondent
The Hamas gunmen who have infiltrated southern Israel are unlikely to remain at large there for long. They will either be killed or captured by the Israel Defense Forces or withdraw by themselves.
The more complicated situation surrounds those Israelis reportedly kidnapped by Hamas and possibly taken into Gaza.
Israel has two options here: mount a special operations rescue mission, or wait it out and negotiate.
Both are fraught with risks.
'I was jogging when the rockets fell, the noise was terrible'
We've been hearing from residents in the Israeli city of Tel Aviv, following the rocket strikes this morning. English author and journalist Gideon Levy told the BBC he believes a settlement with the Palestinians is now out of reach.
"I don't remember Tel Aviv in this situation for many years," he said.
"I just went out on to the streets. The streets are totally empty. Restaurants, cafes everything is locked down and there is a heavy feeling of surprise, of shock and of fear from what is still expected to happen.
"When the first rockets fell, I was still jogging in the park, the noise was terrible", he added.
Israel says it has hit 17 Hamas military compounds
The Israel Defense Forces (IDF) has said it has so far attacked 17 military compounds and four Hamas operational headquarters in a social media post.
The BBC has not verified the claim, but the IDF said earlier it had begun striking targets in the Gaza Strip in response to the earlier barrage of rockets into Israel.
Reaction from around the world
There has been a lot of reaction to the Palestinian attacks in Israel, here's a selection of it:
Iran backs Palestinian attack
We've just received reports saying the advisor to Iran's Supreme leader Ali Khamenei supports the Palestinian attack on Israel.
"We congratulate the Palestinian fighters," Rahim Safavi is quoted as saying by the ISNA news agency.
"We will stand by the Palestinian fighters until the liberation of Palestine and Jerusalem”, he says.
Watch: Tensions and fear near Gaza border
Yolande Knell, the BBC's Middle East correspondent, says there are reports near the Gaza border of Israelis hiding in their homes and calling for the army to be sent there.
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In pictures: Palestinian rocket attacks on Israel
What do we know about reports of hostages?
Graeme Baker
Live reporter
The Israeli television channel Reshet 13 reports that Palestinian militants are holding Israelis captive in the southern town of Ofakim, according to the Reuters news agency.
This is one of the first reports we have from Israeli media on the situation, after widespread claims on social media that Palestinian militants have taken captives.
Unverified videos on social media appear to show Palestinians taking civilians captive and moving them by motorbike into Gaza.
The Israeli authorities have said they will not comment on this and BBC News has so far been unable to verify the reports.
However, a Hamas spokesman told Al Jazeera that "they are not hostages - they are prisoners of war".
Major investigation begins over Israel intelligence failure
Frank Gardner
Security correspondent
Israeli government officials tell me a major investigation is being launched as to how Israeli intelligence failed to see this massive and well coordinated Hamas attack coming.
“It will go on for years,” one says.
BreakingMore than 500 injured - Israel health ministry
Israel's health ministry says at least 545 people have been injured in the Palestinian attacks, according to the Reuters news agency.
At least 22 Israelis have died in the unprecedented attacks, according to emergency services.
Palestinian president hits out at 'occupation forces'
We're hearing more from the Palestinian President, Mahmoud Abbas.
He's been speaking about "the need to provide protection for our people", stressing the right of the Palestinian people to defend themselves against settler "terrorism and occupation forces".
The president was chairing an emergency meeting held in Ramallah with a top officials from the Palestinian Authority.
An unprecedented and unexpected development
Jeremy Bowen
International editor
What’s been happening this morning is without precedent in the more than 15 years since Hamas took control of Gaza.
It comes after months of violence and rising tension between Israelis and Palestinians, although most of that has been concentrated in the West Bank, which is the territory Israel has occupied since 1967, that stretches between Jerusalem and the border with Jordan.
Israel has said it’s been attacked, that effectively Hamas had declared war on Israel, and I would expect that the plans that the Israelis are putting together now will result in a very vigorous response – they’ve said that’s what’s going to happen.
I think there’s a strong chance that will include perhaps putting troops into Gaza.
This situation will move fast and it’s likely to escalate. One big question is what happens now - not just in and around Gaza - but what happens in Jerusalem and the West Bank, where there has been repeated and deepening violence between Palestinian armed groups and Israeli security forces and settlers for months.
An already volatile situation is certain to be exacerbated by these events.
So I think right now for Israelis and Palestinians, this is a thoroughly unexpected and highly worrying and concerning morning.
BreakingAt least 22 Israelis killed - emergency services
Israel's emergency services say at least 22 Israelis have been killed after an unprecedented surprise Palestinian attack.
BreakingPalestinian people have right to defend themselves - Abbas
Palestinian President Mahmoud Abbas says his people have the right to defend themselves against the "terror of settlers and occupation troops", according to the Reuters news agency.
We'll bring you more comments from him shortly.
What we know so far
Things are developing rapidly but this is what we know so far:
UK PM: Israel has right to defend itself
We've just heard from UK Prime Minister Rishi Sunak, who is responding to the unfolding situation.
He says: "I am shocked by this morning's attacks by Hamas terrorists against Israeli citizens."
He continues:
Those by the Gaza strip in a nightmare situation
Yolande Knell
Middle East correspondent, reporting from Jerusalem
This is a time when many Israelis were expecting to enjoy the end of the Jewish Sukkot holiday.
Instead, those living or staying close to the Gaza Strip have found themselves in a nightmare situation with a large-scale infiltration by Palestinian militants. It seems that several Israeli communities near Gaza are being held by Hamas operatives.
On Channel 12, terrified Israelis hiding in safe rooms beg for soldiers to be sent to rescue them. Some say there are militants in their homes.
A woman in Be’eri says her father has been captured. Many of the thousands of people attending an open-air party in southern Israel overnight have been hiding waiting to be rescued.
They say they have been caught up in the exchange of gunfire and report there are casualties. There are Israeli reports that some 60 Hamas militants entered from the Gaza Strip.
Unverified images appear to show Palestinian militants arriving in pickup trucks and by hang-glider as well as riding motorcycles through a hole in the perimeter fence into Israel.
Israeli PM statement in full
"Citizens of Israel. We are at war, not an operation, not an escalation, a war.
"This morning Hamas launched a murderous surprise attack against the state of Israel and its citizens. We've been at it since early morning.
"I convened the heads of the security system, first of all I instructed to cleanse the settlements of the terrorists who had infiltrated - this operation is being carried out during these hours.
"At the same time, I ordered an extensive reserve mobilization and a retaliatory war with a strength and scope that the enemy had never known.
"The enemy will pay a price he has never known. In the meantime, I call on all citizens of Israel to strictly obey the instructions of the army and the instructions of the Home Command.
"We are in a war and we will win it."
Where the situation is unfolding
The map below shows the Gaza Strip and Israel, as well as the Israeli city of Sderot, where Palestinian militants have been seen walking the streets.