But what victory means for each side still remains vague. The Israeli military says the ground offensive is aimed at eliminating militant rocket-launching sites, destroying weapon caches, and pursuing fighters hiding in the crowded coastal strip.
Will it be satisfied if the militants stop firing rockets or if it destroys the hundreds of tunnels to Egypt that make up Hamas's supply line? Some experts say Israel wants to force a more extensive cease-fire with Hamas, compel the creation of an international peacekeeping force in the coastal strip, or destroy the Islamist group altogether.
For Hamas, survival might be victory. It will be lauded across the Arab world if it can hold out against the region's strongest military.
"One of the most important things in this conflict between state and nonstate actors is what is the meaning of victory?" says Eitan Azani, a former Israeli Defense Forces colonel and a deputy director at the Institute for Counter Terrorism at the Interdisciplinary Center in Herzliya. "A lot of people from [Hamas] dying? A collapse? Or most of the operational capability destroyed? This is up for debate. We are in a very complicated situation."
In the 2006 Lebanon war, the Shiite militant group Hezbollah showed the world it could not only survive Israel's superior firepower but that it could confront them on the battlefield. Israel withdrew from the 34-day war with Hezbollah claiming a "divine victory."
So far, Hamas has succeeded in stirring up regional and domestic sympathy under the Israeli pummeling during the first week in the war. But as the fighting continues, the militant group risks seeing its fighting force quickly degraded.
"There may be a push to unseat its hold on Gaza," says Nicolas Pelham, a regional analyst for the International Crisis Group. "It still appears to have retained authority and control in Gaza. There's no internal forces seeking to challenge Hamas."
In the first day of the ground war, Palestinian health authorities reported that 30 Gazans had been killed in the fighting, many of them civilians, according to news wires. An Israeli military spokesperson said that one Israeli soldiers had died in the fighting and three were seriously wounded. Since the operation began on Dec. 27, at least 500 Palestinians have been killed. The United Nations estimates that at least 100 of the dead are civilians.
On Sunday, human rights organizations warned of a burgeoning humanitarian crisis in the Gaza Strip. As a result of the fighting, Gaza City and its main medical center, Shiffa Hospital, have been left without electricity. More than 1 out of every 3 residents are without water and sewage is running in the streets, according to Gisha, an Israeli human rights group.
The ground invasion comes just as international mediators are arriving to the region to begin brokering a cease-fire that is believed to be the seldom-mentioned endgame for both sides.
"The ground action that we began last night, as part of the overall operation, is designed to establish our aspiration to change the security reality in the south," Israeli Prime Minister Ehud Olmert told the Israeli cabinet, according to a statement. "It cannot be that the home front will be subject to attack and a daring, strong and well-trained military does not defend it."
Though Israel has struck at hundreds of targets across the Gaza Strip, it has yet to seriously injure Hamas's fighting force, the Izz ed-Din al Qassam Brigades.
"The main risk for Israel is that it will drag out into a full occupation of the Gaza Strip," says Shlomo Brom, the former head of the army's planning branch. "If we will have very few casualties in this operation, it may lead some to say why don't we topple Hamas?"
Despite the offensive, Palestinian militants were able to fire about 30 rockets into southern Israel Sunday, according to the Israeli army. Since the offensive began, militants have fired hundreds of rockets into Israel, killing three civilians. Israel said its air force hit 15 targets across Gaza on Sunday.
The Qassam wing predicted that the Israeli soldiers would fall prey to the "trap" laid by the Islamist militants.
"The Zionist enemy will be surprised and will regret carrying out such an operation at such a heavy price," the organization said in a statement Saturday. "Our militants are waiting patiently to confront the soldiers face to face."
The start of the ground operation comes on the eve of stepped-up international efforts to prod a cease-fire. Representing the European Union's presidency, Czech Foreign Minister Karel Schwarzenberg, is planning to visit Israel and French President Nicholas Sarkozy is scheduled to arrive Monday.
Much of Israel's firepower has been focused on destroying Gaza tunnels that are a channel for missiles and consumer goods into the strip. Shutting down the free flow of weapons trade over the border figures as one of the major goals of the operation, according to analysts. But it is far from certain who will enforce the closure of the tunnels when Israel leaves.
The nine mile-long Gaza-Egypt frontier has long been crisscrossed by a network of tunnels. The Gaza assault has included several air force sorties in which "bunker buster" bombs were dropped on the area, exploding underground and sending out shockwaves designed to collapse the secret passages. "The issue of rearming is fundamental. We want to prevent Hamas from being rearmed like Hezbollah was after the Lebanon war," a senior Israeli official said.
Palestinians in northern and eastern Gaza said that they can hear the sounds of the Israeli tanks and armored vehicle engines entering Gaza.
Kamel Kafarna, a resident of the northern Gaza village of Beit Hanoun, says Israeli forces are in the fields outside of the village, but haven't entered the residential area. His family of four is running low on flour and wheat, and has run out of cooking gas. But the engineering teacher says he hasn't left his house for three days for fear of getting caught in the crossfire.
"It's crazy. The bullets and rockets are flying over our heads," says Mr. Kafarna, who was reached by phone. "Most of the civilians I believe were killed by mistake. And you don't know if you are going to be the next mistake."