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The plane, a 70-seat ATR 72, crashed on Wednesday evening (local time) near the runway while trying to land on the small island of Penghu, west of Taiwan island, after a typhoon had passed earlier in the day.
The aircraft had 54 passengers and four crew on board. Two of the dead were French, the French foreign ministry said, and 10 people were injured and taken to hospital.
The leaders of rivals China and Taiwan both offered their condolences over the deaths.
Taiwan's civil aviation authorities said the weather had been suitable for flying.
"There were nine flights on the same route between 2 p.m. and 7 p.m. yesterday. Only the TransAsia flight crashed," said Jean Shen, director of the Civil Aeronautics Administration.
"The weather reports showed it was totally OK for landing," she said, adding that authorities were not ruling anything out.
"We can not say for sure what went wrong at this point. The flight safety committee has opened an investigation."
Both black boxes had been found and officials would begin examining them later in the day, she said.
Alison Kao, a TransAsia spokeswoman, said the weather could have been a factor but the airline was not jumping to any conclusions before the investigation.
The aircraft took off from the southern Taiwan city of Kaohsiung, heading for Makong airport in the Penghu islands, but it crashed just short of the runway on its second attempt to land during a thunder storm. The islands are also known as the Pescadores..
No one on the ground was hurt.
Airline seats and life jackets were strewn around the crash site and the roof of a nearby building was destroyed. Victims' families were heading there, officials said.
Taiwan President Ma Ying-jeou said in a statement all of the island's people were grieving.
"Today is a very sad day in the history of Taiwan aviation," Ma said.
China's president, Xi Jinping, who is on a Latin America tour, felt "deeply grieved" after learning of the casualties, the mainland's State Council Taiwan Affairs Office, said in a statement, according to media.
The mainland and Taiwan have been rivals for decades, with the mainland regarding Taiwan as a renegade province, though commercial relations have grown in recent years.
Typhoon Matmo hit Taiwan on Wednesday, bringing heavy rain and strong wind. It later passed the island and headed to China and was downgraded to a tropical storm.
Taiwan has had a poor record for aviation safety over the last two decades, though it has improved recently after the government tightened safety measures.
TransAsia had been involved in eight "incidents" since 2002, including this latest one, according to data on the website of the Aviation Safety Council. It had a fatal accident in 2002 when a cargo plane carrying two pilots crashed into the sea.
TransAsia and bigger rivals, China Airlines and Eva Airways, have been facing pressure from higher energy prices and increasingly popular budget airlines.
TransAsia Airways is a Taiwan-based airline with a fleet of about 23 Airbus and ATR aircraft, operating chiefly short-haul flights on domestic routes as well as to mainland China, Japan, Thailand and Cambodia, among its Asian destinations.