Kim Jong-nam was attacked at about 09:00 (01:00 GMT) on Monday while waiting at the budget terminal of Kuala Lumpur International Airport for a 10:00 flight to Macau,Malaysia's Star newspaper reports, quoting police.
How the attack actually unfolded is still unclear. Malaysian police official Fadzil Ahmat told The Star that Mr Kim had alerted a receptionist, saying "someone had grabbed him from behind and splashed a liquid on his face".
But quoted by Malaysian news agency Bernama, the same official said a woman had come at him from behind and "covered his face with a cloth laced with a liquid". Earlier reports spoke of a "spray" being used or a needle.
His eyes "suffered burns as a result of the liquid", Fadzil Ahmat told Bernama, and he died on the way to hospital in nearby Putrajaya.
"So far there are no suspects, but we have started investigations and are looking at a few possibilities to get leads," Fadzil Ahmat told Reuters news agency separately.
News of Mr Kim's death was not reported until Tuesday. An autopsy was conducted on his body but the results have not yet been released.
Why the delay in identifying him?
South Korean media named the victim early on Tuesday but the Malaysian authorities initially only reported the sudden death of an unnamed North Korean national who had fallen ill at the airport.
Police then released a statement which quoted the victim's travel document identifying him as "Kim Chol", born on 10 June 1970.
Kim Jong-nam was born on 10 May 1971.
Police finally confirmed that the victim was indeed the half-brother of North Korea's leader.
It was not the first time Mr Kim had travelled under an assumed identity: he was caught trying to enter Japan using a false passport in 2001. He told officials he had been planning to visit Tokyo Disneyland.
Why was he flying to China?
Bypassed in favour of his youngest half-brother for succession when their father died in 2011, Kim Jong-nam kept a low profile, spending most of his time overseas in Macau, mainland China and Singapore.
He is said to have enjoyed the slot machines in Macau, a Chinese territory famous for gambling.
How did he fall out of favour?
The Tokyo Disneyland incident is thought to have spoilt his chances of succeeding Kim Jong-il, who died in 2011.
He later spoke out against his family's dynastic control of North Korea and in a 2012 book, was quoted as saying he believed his younger half-brother lacked leadership qualities.