The U.S. is supporting Israel in its push, calling it a reaction to terrorism; Russia has condemned Israel's actions, but is also calling for the release of the captured soldiers.
In the midst of the violence, Japanese Prime Minister Junichiro Koizumi has arrived in the West Bank city of Ramallah, for talks with Palestinian leader Mahmoud Abbas.
Koizumi, who on Wednesday met with Israeli Prime Minister Ehud Olmert in Jerusalem, is pushing for restraint between the Palestinians and Israelis.
In Gaza, the airstrike on the Foreign Ministry badly damaged the building and nearby homes and vehicles, and injured 13 people who lived in the neighborhood.
The Israeli military confirmed it had carried out an airstrike on the ministry, noting that it is "led by Hamas." Israel's offensive is aimed primarily at Hamas, which claimed responsibility for the capture of a soldier in a cross-border raid on June 25.
Israel has rejected demands that it release hundreds of prisoners in exchange for the soldier, Cpl. Gilad Shalit, 19.
Israeli officials on Thursday accused Palestinian Foreign Minister Mahmoud Zahar of having known in advance of the militants' plot to attack an Israeli outpost and seize soldiers.
Zahar "is part of a leadership that is involved in a very tangible way in terrorism and in violence," said Israeli Foreign Ministry spokesman Mark Regev. "He had knowledge of the recent hostage taking, and he is part of a leadership that has orchestrated ... countless missile attacks against Israeli urban areas."
Foreign Ministry spokesman Taher al-Nunu accused Israel of carrying out "organized terrorism that targets all the Palestinian people and aims to exterminate all government institutions, one after another, to prevent them from carrying out their duties." He said no one was in the building when the plane struck after 1:30 a.m. Thursday.
In other recent developments:
In Lebanon, Israel's military campaign is the heaviest in 24 years.
The Israeli attacks in south Lebanon alone killed 35 civilians and wounded 52 more, Lebanese security officials said. A family of 10 and another family of seven were killed in their homes in the village of Dweir near Nabatiyeh, the officials said, speaking on condition of anonymity as they were not authorized to speak to the press.
Israeli Defense Minister Amir Peretz said his forces would not allow Hezbollah guerrillas to occupy positions along the southern Lebanese border.
"If the government of Lebanon fails to deploy its forces, as is expected of a sovereign government, we shall not allow Hezbollah forces to remain any further on the borders of the state of Israel," Peretz said.
Lebanon has long refused to deploy more than a limited number of security forces in the border region, saying it is not in business of protecting Israel's northern border.
"The government wants to change the rules of the game in Lebanon and make the Lebanese government understand that it is responsible for what happens in Lebanon," Israeli Agriculture Minister Shalom Simchon told Israel Radio.
Israeli fighter bombers carried out their biggest offensive in Lebanon since Israel's 1982 invasion after the Cabinet approved a forceful response to Hezbollah's daring raid across the border on Wednesday to capture the Israeli soldiers. Three Israeli soldiers were killed in that raid, and five others died during fighting since.
Air force Maj. Gen. Amir Eshel said the campaign is likely Israel's largest ever in Lebanon "if you measure it in number of targets hit in one night, the complexity of the strikes." The last major air, ground and sea offensive against Lebanon was in 1996 when about 150 Lebanese civilians were killed.
Trying to avoid devastating Israeli bombardment of the nation's infrastructure, the Lebanese government insisted Wednesday it had no prior knowledge of the Hezbollah operation, did not condone it and bore no responsibility for it. The Cabinet, which includes two Hezbollah ministers, urged the U.N. Security Council to intervene.
Senior Israeli military officials said Israel warned the Lebanese government that it plans to strike offices and homes of Hezbollah leaders in the southern suburbs of Beirut.
An airstrike also hit deep inside eastern Lebanon, striking a civic center attached to a Shiite Muslim mosque near the town of Baalbek, as well as a transmission antenna for Al-Manar, witnesses reported. The group's broadcasts stopped in the area.
After dawn, warplanes struck the runways of Beirut's international airport, located by the seaside in the Lebanese capital's Hezbollah-controlled southern suburbs, forcing its closure and the diversion of incoming flights to Cyprus.
Travelers to and from Beirut were stranded all over the region and beyond.
The Israeli military said it struck the airport because it is "a central hub for the transfer of weapons and supplies to the Hezbollah terrorist organization."
It was the first time since Israel's 1982 invasion of Lebanon and occupation of Beirut that the airport in south Beirut was hit by Israel. The Israelis in 1968 sent commandos to Beirut airport, blowing up 13 passenger planes on the runway in retaliation for Arab militants firing on an Israeli airliner in Athens.
In its overnight attacks, Israeli aircraft and artillery targeted roads and bridges as well as Hezbollah positions and houses of guerrilla members and leaders. Meanwhile, helicopter gunships and jet fighters scoured southern Lebanon for guerrillas launching rockets into northern Israel.
Hezbollah fired volleys of rockets at the Israeli town of Nahariya, and said that in some of its attacks it was using a rocket called "Thunder 1" for the first time. The missile appeared to be more advanced than the inaccurate Katyusha which has been the standard Hezbollah rocket for years.
The Israeli army said several rockets had landed more than 20 kilometers (12 miles) south of the border, showing that Hezbollah has managed to extend its missiles' range.
A 40-year-old Israeli woman was killed when her home in Nahariya was struck by a rocket. At Nahariya Hospital, patients were moved to secure rooms on lower floors. Nahariya's Mayor Jackie Sabag said the whole town had been shut down and urged residents to stay in underground shelters.
Hezbollah's TV station reported Thursday that guerrillas fired Katyusha rockets at the Israeli border town of Kiryat Shmona, targeting an airstrip in the upper Galilee panhandle.
Hezbollah has declared it has over 10,000 rockets and has in the past struck northern Israeli communities in retaliation for attacks against Lebanese civilians.