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The explosives went off within 75 minutes of each other across several districts of India's financial capital.
The first bomb went off at 1325 local time at the Bombay Stock Exchange. The blast occurred in the basement garage. It blew up more than 30 cars and shattered windows.
Some brokers and investors are reported to have been trampled to death in the stampede.
Mr Mayya, chief executive of the Bombay Stock Exchange, said: "There was blood everywhere and people were rushing to get out."
He described how there was chaos in the lower floors of the building and on the streets outside as office workers tried to flee high-rise buildings and make their way to the railway and bus stations.
Police have cordoned off the area around the exchange, which is littered with shards of glass and metal rods.
Up to 20 bodies have been removed from the headquarters of Air India, which was also bombed.
Witnesses said glass, furniture and bodies had been flung into the road by the force of the explosion at the Air India building, one of Bombay's high-rise landmarks.
The airline HQ was among a cluster of commercial buildings in Nariman Point that were attacked. The Bank of Oman was destroyed by a bomb.
Other explosions were reported at government offices, banks, cinemas, bazaars, two hospitals, a university and several hotels. A bus was also torn apart by a bomb.
A police spokesman said all the bombs were set off by timing devices and were made of high quality plastic explosives and the Stock Exchange device appeared to be a car bomb.
The spokesman said no-one had stepped forward to admit carrying out the attacks but the blasts appeared to have been timed to paralyse business and trading and spread fear among Bombay residents.
Government officials said most of the bombs were in vehicles but several were in unoccupied hotel rooms.
In New Delhi, the Indian Home Affairs Minister, SB Chavan, said the attacks were part of "an international conspiracy".
No arrests have been made - but explosive experts have been flown in from Delhi to analyse the devices used in the attacks.
The number of dead in Bombay is expected to rise as police and firefighters find more charred bodies from the bombed areas of the city.
Hospital officials said they expected the final number of dead to increase as many people had suffered serious injuries.
England's hockey team, in Bombay for an international tournament, escaped serious injury when one bomb went off at the Searock Sheraton Hotel, where they were staying. Some players were cut by flying glass.
It later emerged that at least 250 people had died in 15 bombings, 1,100 had been injured and the attacks had caused $10m in damage.
The bombings were seen as a retaliation for anti-Muslim riots that left hundreds dead.
The Mumbai (Bombay) bombings case is one of India's longest-running trials.
A special court was set up to hear the case and regular hearings have taken place over the years, with more than 600 witnesses questioned.
Bombay, which is home to 12.5 million people, has been the scene of violence since Indian independence in 1947.
At the time of the 1993 bombings, Bombay was still recovering from a wave of Hindu-Muslim fighting in January of that year, which left more than 500 dead and led to the flight of around 40,000 people from the city.
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