Officials are fearful that dozens have died, though Taiwan's Transportation Minister Yeh Kuang-shih told reporters there were no casualties on the ground.
Injured passengers have been rushed to Penghu Hospital, and TransAsia Airways has established an emergency response center, according to a statement issued by the airline.
The Penghu Islands are off the west coast of the main Taiwanese island.
The president of TransAsia Airways, Chooi Yee-choong, appeared briefly at a press conference and bowed in front of news cameras. He choked up as he expressed his sorrow to passengers' families and the public for the tragedy. "I sincerely apologize," he said.
There were 54 passengers and four crew aboard, said Jean Shen, the director-general of the Civil Aeronautics Administration.
Before Flight GE222 took off from Kaohsiung, Taiwan, it had been delayed due to conditions related to a typhoon, the airline said.
"TransAsia Airways is exhausting all means to assist passengers, victims and families" and working with investigators, its statement read.
The transportation minister said that two French nationals were among those on board, and French authorities have been notified.
One of the plane's "black box" data recorders was recovered and investigators will examine the crash site Thursday, the minister said.
CNN is working on getting details about the crash, and has spoken with Taiwan's Civil Aeronautics Administration, which is saying only that dozens are missing and others are hospitalized. Officials caution that numbers could change.
The plane crashed near Magong Airport at about 7 p.m., according to CNA. Witnesses told ETTV that they saw homes on fire.
The cause of the crash is unknown so far.
Some media reports said strong winds from Typhoon Matmo, which hit Taiwan early Wednesday, forced the plane to attempt a crash landing.
Shen told reporters that visibility at Magong Airport at the time of the plane's attempted landing was about 1,600 meters (1 mile) and considered acceptable for landing.