Israel launched the raids after Hezbollah, the Lebanese political and military organisation, attacked northern Israel on Wednesday killing eight soldiers and abducting two.
Hezbollah responded to Thursday's airstrikes with further rocket attacks against Israeli towns, killing one woman in the city of Nahariya.
Beirut's new Rafiq al-Hariri airport was closed after Israeli jets bombed its runways on Thursday morning.
The Israeli military confirmed that its aircraft carried out attacks on the airport.
"The airport is used as a central hub for the transfer of weapons and supplies to the Hezbollah terrorist organisation," an Israeli army spokesman said.
The airport's three runways have all been damaged and flights will not resume for several days. Civilian flights to Beirut have been diverted to Cyprus.
Israel also announced that its navy had entered Lebanese territorial waters to blockade Lebanon's ports.
It also bombed bridges linking south Lebanon to the rest of the country and hit the Beirut offices of Hezbollah's Al-Manar television station, Aljazeera's correspondent in Beirut said.
The attack killed one Hezbollah fighter. Three more employees were injured in another raid on the channel's offices in Beirut's southern suburbs.
In southern Lebanon, 10 members of the Adil Akashi family and seven members from the family of Muneer al-Zain family were among those killed in the Israeli bombings, an Aljazeera correspondent said.
Hezbollah said it had responded to the airstrikes by firing 60 rockets at Israeli city of Nahariya, just south of the Lebanese border.
The grandmother of slain Israeli
soldier Shani Turgeman grieves
Israel's Magen David Adom ambulance service said that one 40-year-old woman was killed when a rocket fired from Lebanon hit her house in Nahariya.
The Israeli army also said that Hezbollah had fired a number of rockets and mortars at other areas in northern Israel.
One rocket landed in the Israeli town of Zarit, wounding one civilian. Others injured 11 civilians in the town of Safed.
Also on Thursday, Hezbollah threatened to strike the northern Israeli city of Haifa and surrounding areas if Beirut and its southern suburbs are hit by the Israeli military.
"The Islamic Resistance declares that it will shell the city of Haifa and surrounding areas if the southern suburbs and Beirut are the target of a direct Israeli aggression," it said in a statement.
On Wednesday, the two Israeli soldiers were seized at about 9am (0600 GMT), across the border from Aita al-Shaab, 15km from the Mediterranean coast. Prisoner swap wanted
Three Israeli soldiers died in the raid, while four more were killed when their tank ran over a landmine. The eighth soldier died while helping to recover the tank, the Israeli army said.
Israeli officials said the air attacks aimed to exert pressure on the Lebanese government to prevent Hezbollah launching cross-border attacks.
The Israeli prime minister described Wednesday's attack as an act of war by Lebanon and promised a "severe response". The government approved miltary action against Hezbollah after an emergency cabinet meeting.
Sayyed Hassan Nasrallah, leader of Hezbollah, said the soldiers had been seized to push Israel to release prisoners.
Nasrallah: Only feasible path to
free detainees from Israeli jails
"What we did ... is the only feasible path to free detainees from Israeli jails," he told a news conference in Beirut.
But Israel said on Thursday that it would not swap prisoners held in its jails for the two captured soldiers.
Gideon Meir, an Israeli foreign ministry official, said: "You don't negotiate with terror organisations."
Nasrallah said Hezbollah had repelled an Israeli force that tried to enter Lebanon to find the captured soldiers.
Footage on Al-Manar television showed a smouldering Israeli jeep. It also showed smoke rising from an Israeli border post.
Hezbollah supporters set off fire crackers and distributed sweets in the streets of Beirut in celebration of the attack.
Mothers of Lebanese prisoners in Israeli jails expressed hope to Aljazeera that the Hezbollah attack could increase the possibility of their relatives being freed, while in Gaza the families of some Palestinian prisoners also welcomed the attack.
Israel bombarded Lebanon from
land, sea and air
Israeli troops have not struck so far into Lebanon since they withdrew from the southern border area in 2000.
Hezbollah's Shia fighters had waged an 18-year guerilla campaign against them following the Israeli invasion of Lebanon in 1982.