According to the Taliban spokesman Zabiullah Mujahid 20 armed militants, including suicide bombers, entered the capital city to target the presidential palace and other government buildings.
Heavy firing took place around the areas near presidential palace, the central bank and the ministry of justice.
The areas in Central Kabul has been sealed off by the police.
Interior Ministry spokesman Zemari Bashary in media report said that one group of militants have entered the supermarket after hurling grenades to frighten off shoppers.
"The area is surrounded by security forces. A gun battle is going on," he said.
President Hamid Karzai and some of the cabinet ministers where being sworn in, when the attack took place.