The ferry, carrying mainly school students, was travelling from the port of Incheon, in the north-west, to the southern resort island of Jeju.
A major rescue effort is under way, involving dozens of ships and helicopters. Those brought to safety were taken to a nearby island.
Four people are now said to have died and dozens of others have been hurt.
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Richard Westcott BBC Transport correspondent Domestic ferries across some parts of Asia have a bad safety record, especially in countries such as the Philippines and Indonesia where ships can be overcrowded and poorly maintained. But in recent years South Korea has been one of the exceptions.
This particular ferry was built in Japan, who make some of the best ships in the world. It wasn't anywhere near full and it was travelling a well-worn route in reasonably calm seas. The speed with which it flipped over and sank is a major concern.
Passengers described a loud thud, which might suggest it hit something hard. It doesn't matter how well built the ship is, collisions can sink vessels very quickly.
Survivors have also complained that they were told to stay put even though the ferry was tipping over. That's not as surprising as you might think. The basic assumption is always that the ship is the best lifeboat. That it's normally safer to stay on board than brave the water.
All of these split second evacuation decisions and the design of the ship, will be analysed by investigators.
South Korean officials had earlier said that 368 people had been plucked to safety, but later said there had been a counting error.They have now revised down the number rescued to 164.
Images showed the ferry listing at a severe angle and then later almost completely submerged, with only a small part of its hull visible. It sank within two hours of sending a distress signal, reports said.
There are fears this could turn out to be South Korea's biggest maritime disaster for more than 20 years, says the BBC's Martin Patience in Seoul.
Cause unclearSeveral coast guard, military and commercial vessels were involved in the rescue effort, which unfolded rapidly on Wednesday morning.
Pictures from the scene showed rescue teams balanced on the sinking hull pulling teenagers from cabin windows. Some of their classmates jumped into the sea as the ship went down.
Reports suggest some of those rescued were picked up by nearby commercial vessels. The US Navy has sent a ship to assist the search.
Continue reading the main storyPassengers on board: 447
Maximum capacity: 900
Length: 146 metres
Built: 1994
Military and civilian ships and helicopters have been searching for survivors 
Reports said the ship capsized and sank within a period of two hours Teams of navy divers were now searching the scene for those unaccounted for, officials said, but the work was challenging. "There is so much mud in the sea water and the visibility is very low," said Lee Gyeong-og, vice-minister of security and public administration.
One body, of a female crew member, had been recovered from the ship, the coastguard said. Another person, a male high school student, died after being rescued.
Continue reading the main story 1970: Sinking of passenger vessel Namyoung leaves 323 dead1993: Sinking of passenger vessel Seohae Ferry leaves 292 dead2007: Sinking of freighter Eastern Bright leaving 14 sailors missing2009: Sinking of cargo ship Orchid Pia after a collision leaves 16 sailors missingSource: Yonhap
One student told local media her friends became trapped."Currently, I am in the middle of being rescued. At the time, the ship was turning on its side, and none of us were moving as we were told not to move as it was dangerous," the unnamed student said.
"So, I am not well aware of the situation, but I am told that my friends and other friends could not escape as the passage was blocked. It seems that there are many students who could not get out as the passage was blocked by water."
It is not yet clear what caused the incident, but witnesses described hearing an impact, before the ship listed and quickly sank.
One passenger told the YTN news channel: "We heard a big thumping sound and the boat stopped."
Teams have brought rescued passengers to shore - at least 13 are reported to be hurt 
Flares light up the sky as the search for missing passengers continues through the night 
Relatives of the missing are waiting to hear of the fate of their loved ones "The boat is tilting and we have to hold on to something to stay seated," the passenger said.
Another passenger said the ship was "shaking and tilting", with people tripping and bumping into each other.
South Korean President Park Geun-hye has expressed sadness over the incident, saying it was "truly tragic" that students on a field trip were involved in "such an unfortunate accident".
Kim Young-boong, an official from the company which owns the ferry, has apologised.
Survivor: "There was an announcement telling us to sit still, but the ferry was already sinking""I would like to say sorry to the passengers, which include a number of students and their parents, and promise that our company will do its best to minimise loss of life. We are sorry," he said, according to the AP news agency.
Weather conditions were described as fine. Yonhap news agency said that the ferry sank at a depth of 30m (90ft).
Continue reading the main story Ferry - named Sewol - was travelling from Incheon to Jeju Island, a route it serves twice a weekThe 146m-long vessel has a capacity of about 900 people but was carrying about 470 peoplePassengers include about 330 students from Danwon High School in Ansan, a suburb of SeoulNews agencies said the ferry had sent out a distress signal about 20km (12 miles) off the island of Byungpoong at about 09:00 local time (00:00 GMT)."We will try to determine the cause of the accident after rescue operations are over," said Lee Gyeong-og.
Many of the passengers were students from a high school in a suburb of Seoul heading off on a four-day field trip to Jeju.
Angry parents have gathered at the school in Incheon to demand answers, reports the BBC's Lucy Williamson in Seoul.
Earlier reports put the number of passengers on the ferry at about 350. The vessel is reported to have a capacity of up to 900 people.
Lee Gyeong-og, the vice-minister of security and public administration, was quoted by the AP news agency as saying that 30 crew members, 325 high school students, 15 school teachers and 89 non-student passengers were aboard the ship.
Are you in the area? Do you have any information you would like to share? Please send us your comments. You can email us at haveyoursay@bbc.co.uk using the subject line 'South Korea ferry'.
