"We're going to catch them dead or alive," R R Patil, home minister for Maharashtra state, told reporters.
"An attack on Mumbai is an attack on the rest of the country."
Six foreigners are among the dead, police have said.
Pollice loudspeakers have announced a curfew around the areas where the attacks have taken place.
Reports say that the attackers had first made their way to the Taj hotel on an inflatable boat.
"They told everybody to stop and put their hands up and asked if there were any British or Americans … My friend said to me, 'don't be a hero, don't say you are British.'"
- Alex Chamberlain, British guest at the Oberoi Trident who fled via a fire escape
"That was, without doubt, the worst experience of my entire life … We could hear the army coming through the hotel ... The firemen broke the windows of the room and we climbed down the ladder."
- Female guest, Taj Mahal hotel
"They wore black T-shirts and blue jeans. They were carrying big guns … They just fired randomly at people and then ran away. In seconds, people fell to the ground."
- Nasim Inam, at Chhatrapati Shivaji railway terminus
"As I stepped inside the lobby gunshots started to go off ... There were people getting shot in the corridor. There was someone dead outside the bathroom."
- Brooke Satchwell, an Australian television actress, who hid in a cupboard in the Taj Mahal hotel
"We had literally just ordered and then it seemed like firecrackers – people were screaming … I turned around and she was crawling out the door because she couldn't walk."
- David Coker, an Australian student, who was shot along with his girlfriend at the Cafe Leopold restaurant
"We have been told that there were 12 people [on the boat]; they split up into smaller groups and then went off to the various locations where the attacks took place," James Wright, reporting for Al Jazeera, said.
Between 100 and 200 people are believed to be still inside the Taj hotel, Patil said.
"We cannot give you the exact figure as many people have locked themselves inside their rooms," he told reporters.
"There could be 10 to 12 terrorists inside the hotel. There are no negotiations with the terrorists."
Witnesses at both hotels said the attackers singled out British and American citizens.
"They kept shouting: 'Who has US or UK passports?'" said Ashok Patel, a British citizen who fled from the Taj Mahal.
Several European legislators, visiting Mumbai ahead of a European Union-India summit, are reportedly among those who have barricaded themselves inside the hotel.
It was also unclear where the hostages were in the Taj Mahal, which is divided into an older wing, part of which was in flames, and a modern tower that was not on fire.
Indian police, said eight locations were targeted in what they called terrorist attacks, including the two hotels.
Other hotels, the crowded Chhatrapati Shivaji train station, Leopold's restaurant and the police headquarters in southern Mumbai have also been hit.
A Jewish rabbi and his family are also thought to be under threat after gunmen stormed the building in which they were staying.
"The name of the place is Chabad house in South Mumbai. I hear commandos are storming the apartment block, which is a four-storey building," Jonathan Solomon, the chairman of the Indian Jewish Federation, said.
"A rabbi is in there with his family ... I do not know the number of gunmen in there. I don't know how many family members are in there."
'Shock' at attacks
Al Jazeera's Riz Khan, reporting from Mumbai, said the targeted buildings are popular not only with tourists but local elites as well, and were clearly chosen by the attackers to give their assaults maximum exposure.
|A popular restaurant, a busy railway station and a police station were also hit [AFP]
Shai Venkatraman, a correspondent with the private New Delhi Television channel, said that Mumbai was in a state of shock following the co-ordinated attacks.
"Mumbai is not new to terror but this has really shocked people. There is a stunned silence," she told Al Jazeera.
"This has really damaged Mumbai's reputation, even though it is a city that is usually quick to come back. But this time, the attack has really shaken the people.
"The situation at the Taj hotel is especially bad. There are reports that there are about four terrorists there and we are hearing that about 20 to 25 people are trapped at the hotel.
"There are no negotiations going on. At the moment, there are just exchanges of gunfire."
Group claims attacks
A little known group calling itself the Deccan Mujahedeen claimed responsibility for the attack in emails to news organisations, but Mahan Abedin, an insurgency analyst, told Al Jazeera he had never heard of the group.
"At this stage, that name does not necessarily mean that much," he said.
"It is very tempting to point the finger at Muslim militants, and indeed they have been the main culprits in the pattern of attacks", but cautioned that "one has to be careful before pointing the finger," Abedin said.
"We have seen an increase in recent years in indigenous Indian Muslim organisations beginning to take a violent stance towards the Indian state and sections of the Indian society, particularly the commercial elite of places like Mumbai, in order to highlight, they would say, the sheer inequality of life in India.
"There is a middle class of around 100 million who live very well but 800 million-plus people live in miserable conditions," he said.
Anti-terror chief killed
At least three senior Indian police officers, including Hemant Karkare, the chief of the city's anti-terrorism squad, were among those killed, police official A N Roy said.
|The co-ordinated attacks are the most severe to have hit Mumbai in months [AFP]
Maharashtra's chief minister said 11 police officers were killed in all.
The motive for the attacks was not immediately clear, but India has witnessed a series of co-ordinated attacks in recent months.
A little-known group called the Islamic Security Force-Indian Mujahedeen, claimed responsibility for a series of blasts last month in which 80 people died in the northeast state of Assam.
A total of 12 explosions shook the northeastern state, six of them ripping through crowded areas in the main city of Guwahati.
Six weeks earlier, the capital New Delhi was hit by a series of bombs in crowded markets that left more than 20 people dead.
Those attacks were claimed by a group calling itself the Indian Mujahedeen.