Israel's leaders have declared that Hamas will be wiped off the face of the Earth and Gaza will never go back to what it was.
"We have set two goals for the war: to eliminate Hamas by destroying its military and governing capabilities and to do everything possible to bring our hostages at home," announced Prime Minister Benjamin Netanyahu, as Israel expanded its ground operations in Gaza.
Operation Swords of Iron began in response to the killing of more than 1,400 people by Hamas gunmen in a brutal attack on Israeli communities, and Israel's chief of staff has said the military are fighting "for our right and the right of future generations to live in safety and prosperity in our homeland".
The goal of ending Hamas rule has been endorsed by US Secretary of State Antony Blinken.
Once Israel is satisfied Hamas is defeated, its stated aim is to disengage from Gaza completely. Defence Minister Yoav Gallant speaks of a "new security regime" there, with no Israeli responsibility for daily life.
Israel's aims appear far more ambitious than anything the military has planned in Gaza before and could last months. But are they realistic, and how can its commanders possibly fulfil them?
Entering the Gaza Strip involves house-to-house urban fighting and carries immense risks to a civilian population of more than two million. Officials in Hamas-run Gaza say more than 8,700 people have already been killed, and hundreds of thousands more have fled their homes.
The Israel Defense Forces (IDF) have the added task of rescuing 239 hostages, held in unknown locations across Gaza.
"I don't think Israel can dismantle every Hamas member, because it's an idea of extremist Islam," says military analyst Amir Bar Shalom of Israel's Army Radio. "But you can weaken it as much as you can so it has no operational capabilities."
Weakening it might be a more realistic objective than dismantling it entirely. Israel has already fought four wars with Hamas, and every attempt to halt its rocket attacks has failed.
An IDF spokesman said the main aim was that Hamas should no longer have the military capacity to "threaten or kill Israeli civilians".
Michael Milstein, head of the Palestinian studies forum of Tel Aviv University, agrees that destroying Hamas would be highly complicated. He says it would be pretentious to believe you could uproot the idea that embodies Hamas - an offshoot of the Muslim Brotherhood that has influenced Islamist movements around the world.
Quite apart from the 25,000-plus strength of Hamas's military wing, it has 80-90,000 more members who are part of its social welfare infrastructure, or Dawa, he says.
Ground invasion fraught with risk
The military operation is at the mercy of several factors that could derail it.
Hamas's armed wing, the Izzedine al-Qassam Brigades, will have prepared for an Israeli offensive and can use its notorious and extensive network of tunnels to attack Israeli forces.
In 2014, Israeli infantry battalions suffered heavy losses from anti-tank mines, snipers and ambushes, while hundreds of civilians died in fighting in a northern neighbourhood of Gaza City.
This time too the civilian death toll has soared since ground operations expanded on 27 October, and the IDF has lost increasing numbers of soldiers in combat.
Israelis have been warned to prepare for a long war, and a record 360,000 reservists have reported for duty.
The question is how long Israel can continue its campaign as international pressure grows for a ceasefire, with supplies of water, power and fuel cut off, and UN warnings of an unfolding human catastrophe.
"The government and military feel they have the backing of the international community - at least Western leaders. The philosophy is 'let's mobilise, we have plenty of time'," says Yossi Melman, one of Israel's leading security and intelligence journalists.
But sooner or later he believes Israel's allies will step in if they see images of people starving. Pressure will also mount as civilian casualties continue to rise.
"It's very complicated because it needs time and the US administration won't let you stay in Gaza for a year or two," says Michael Milstein.
Saving the hostages
Many of the hostages are Israelis, but there are also a large number of foreign citizens and dual nationals among them, meaning several other governments - including the US, France and the UK - have a stake in this operation and their safe release.
President Isaac Herzog has said it is Israel's top priority to bring them home. But French strategic specialist Col Michel Goya believes the IDF had a straightforward choice, either sparing the hostages' lives, or going in to "do Hamas as much harm as possible".
Heart-rending appeals from families of those in Hamas captivity are ramping up pressure on Israel's leaders, and Hamas's political leader in Gaza, Yahya Sinwar, has said they are ready to swap all the hostages for the thousands of Palestinian prisoners in Israeli jails.
That kind of deal is unlikely.
In 2011, Israel exchanged more than 1,000 prisoners for the release of a soldier, Gilad Shalit, held by Hamas for five years. Yahya Sinwar was among those freed, and is now their top man in Gaza.
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Neighbours watching closely
What could also affect the duration and outcome of a ground offensive is how Israel's neighbours react.
Egypt's Rafah border crossing with Gaza has become a humanitarian focal point for supplies getting into Gaza. Hundreds of foreign passport holders and some of the wounded have been allowed to leave but more remain.
"The more Gazans suffer following the Israeli military campaign, the more pressure Egypt will face, to appear as if it has not turned its back on the Palestinians," says Ofir Winter of Israel's Institute for National Security Studies.
But that will not stretch to Cairo allowing a mass crossing of Gazans into northern Sinai. President Abdul Fattah al-Sisi has warned that any attempt to move Gazans into the Sinai peninsula would prompt Egyptians to "go out and protest in their millions".
Jordan's King Abdullah has also talked of a "red line" for any possible attempt at pushing Palestinian refugees out of Gaza: "No refugees in Jordan, no refugees in Egypt," he warned.
Israel's northern border with Lebanon is also under close scrutiny.
There have already been regular, deadly cross-border attacks involving Islamist militant group Hezbollah, but although communities on both sides have been evacuated, so far the violence does not amount to a new front against Israel.
Iran, Hezbollah's main sponsor, is already threatening to launch "new fronts" against Israel. They were the focus of US President Joe Biden's warning, when he said: "To any country, any organisation, anyone thinking of taking advantage of this situation, I have one word: Don't!"
Two US aircraft carriers, the USS Gerald Ford and USS Eisenhower, have been sent to the eastern Mediterranean to emphasise that message, and 2,000 US troops have been placed on alert to respond to unfolding events.
What is Israel's endgame for Gaza?
If Hamas were to be significantly weakened, the question is what would go in its place.
Israel pulled its army and thousands of settlers out of the Gaza Strip in 2005 and will have no intention to return as an occupying force. US President Joe Biden says "it would be a big mistake".
A power vacuum would also create very serious risks and Mr Milstein warns the risk of solving one problem is in finding 10 new ones.
Ofir Winter believes a shift in power could potentially pave the way for the gradual return of the Palestinian Authority (PA), kicked out from Gaza by Hamas in 2007.
The PA currently controls parts of the West Bank, but is weak there, and persuading it to return to Gaza would be highly complicated.
Antony Blinken has said it would make "most sense... for an effective and revitalised" PA to run Gaza, although a number of other countries in the region could take part in "temporary arrangements" that might involve international agencies.
The United Nations temporarily ran Kosovo after Serbian forces withdrew in 1999, but the UN is widely mistrusted in Israel.
Another option might be to create an administration, run by Gaza's mayors, tribes, clans and non-governmental organisations, with involvement from Egypt and the US, the PA and other Arab states, says Michael Milstein.
Egypt's president has himself shown no interest in controlling Gaza but has pointed out that if a "demilitarised Palestinian state had been created long ago in negotiations, there would not be a war now".
Gaza's devastated infrastructure will ultimately have to be rebuilt in the way it was after earlier wars.
Israel will want even tighter restrictions on "dual-use goods" entering Gaza that could have a military and civilian purpose. And there have been calls for an expanded buffer zone along the fence with Gaza to provide greater protection for Israeli communities.
Whatever the outcome of the war, Israel will want to ensure a similar attack never happens again.