Casualties reported in Kabul
Several people died in the bomb raids on Kabul, witnesses told Reuters news agency.
"People have seen some bodies from the attacks. There are several deaths," the agency quoted a witness as saying - but it wasn't clear if the casualties were civilian.
Associated Press reports that the Taliban stronghold of Kandahar - a shabby city of rocket-gouged streets and bullet-scarred homes - appears to have taken the brunt of the attacks.
The assault against the southern Afghan city came in at least three waves, according to Taliban sources, who spoke to the Associated Press by telephone on condition of anonymity.
The first was a punishing attack on the airport, ironically built by the United States as a refuelling stop for long distance aircraft between Europe and India before the long-haul Boeing 747s were introduced.
In the second wave, the Taliban's military headquarters and the home of Taliban supreme leader Mullah Mohammed Omar were struck. The Taliban ambassador in Islamabad, Pakistan, Abdul Salam Zaeef, said Omar was alive after the attack.
Zaeef did not say whether Omar was at his residence when the attack began.
During the Soviet occupation of Afghanistan that began in 1979, Kandahar airport served as a base for airstrikes against the US-backed Islamic resistance movement.
Today the airport complex includes 300 houses built in 1996 for fighters from Osama bin Laden's al-Qaeda terrorist network. Bin Laden is the main suspect in the September 11 terror attacks on the United States.
The refusal by the Taliban to hand over bin Laden prompted the attack on Afghanistan - a country reminiscent of the 7th century, both in its infrastructure and in the thinking of the Taliban regime, who rule most of it.
The sophistication of the force arrayed against the Taliban tribesmen was brought to bear when the second wave struck the city, according to the sources.
This hit was precise. They said missiles slammed into the Taliban's military headquarters in the heart of the city. The mud-brick homes and high-walled compounds that line the same sewer-lined road as the military headquarters apparently escaped damage.
Last week, several Kandahar residents, who had fled to neighbouring Pakistan, said that the heavily armed Taliban, who routinely could be seen loitering outside the military headquarters, had disappeared.
Inside the building only a few people could be seen and they were mostly Arab members of al-Qaeda, they said.
Witnesses in Kandahar reported seeing black smoke billowing from the home of Mullah Omar. Omar moved to the high-walled compound about 15 kilometres outside Kandahar last year after a powerful explosion outside his former residence in the city.
That blast killed 42 people, including several of his guards. The anti-Taliban northern alliance was blamed for the explosion.
The United States said the target of the strikes was the Taliban leadership, which harbours bin Laden, and the al-Qaeda terrorist network that operates camps throughout the country.
In the eastern Afghan city of Jalalabad, there were reports of three explosions including one south of the city in Farmada, where hundreds of Arab fighters once lived.
Afghans who travel freely across the border say that most of the camps used by militants - mostly Arabs, Pakistanis, Uzbeks and Chechens - were abandoned after the September 11 attack in the United States.
Rishkore, a camp on the southern edge of Kabul, was also apparently targeted in the US attack. It was deserted months before the September 11 attack, according to residents in the capital.
However, the infrastructure remained, including houses, offices and training facilities.
Destroying the al-Qaeda training camps in the rugged mountain ranges that crisscross Afghanistan will be more difficult, according Taliban and other sources familiar with them.
"They have camps in every province," said one senior Taliban commander, who spoke on condition of anonymity.
Some former rebel commanders believe the success of the US-led campaign will depend on attacks that are quick, precise and target only the Taliban and bin Laden.
Abdul Haq, a former Kabul area commander during the war against the Soviets, said precision is the key.
Haq is likely to play a major role in forming a future government to replace the Taliban.
"First of all, honestly, the attack should be specific to areas where Arabs are. It shouldn't be long and they shouldn't bomb the cities," he said in a telephone interview today shortly before the assault began.