According to The Guardian  newspaper, the aircraft lost radar contact in Vietnamese airspace.
The airline has since posted several times on its Facebook page providing public updates about the situation, including a statement  by its CEO Ahmad Jauhari Yahya:
We deeply regret that we have lost all contacts with flight MH370 which departed Kuala Lumpur at 12.41 am earlier this morning bound for Beijing.
The aircraft was scheduled to land at Beijing International Airport at 6.30am local Beijing time.
Subang Air Traffic Control reported that it lost contact at 2.40am (local Malaysia time) today.
Flight MH370 was operated on a Boeing B777-200 aircraft.
The flight was carrying a total number of 239 passengers and crew – comprising 227 passengers (including 2 infants), 12 crew members. The passengers were of 13 different nationalities.
Malaysia Airlines is currently working with the authorities who have activated their Search and Rescue team to locate the aircraft.
Our team is currently calling the next-of-kin of passengers and crew.
Another press release  by the airline identified the pilot as Captain Zaharie Ahmad Shah, who has 18,365 flying hours under his belt.
On Twitter, many have expressed concern and worry:
Flown Malaysia Airlines @MAS  many times, including one of their 777-200s from KL to Shanghai. Thoughts with all at this hard time.
— Peter McGinley (@PeterMcGinley) March 8, 2014 
The Malaysia Airlines’ flight that disappeared is exactly why I'm terrified of flying — Jonathan Woelk (@jonathanwoelk) March 8, 2014 
Fingers crossed the Malaysia Airlines plane did not crash and the 239 people aboard are safe and sound — C0l\l0R. (@F_ckOhioSt) March 8, 2014 
Very shocking news for aviation industry and those onboard! BBC News – Malaysia Airlines loses contact with plane flying to Beijing — Stephen Chimalo (@SChimalo) March 8, 2014 
The incident even prompted a response from the Malaysian Prime Minister, Najib Razak, on Twitter:
My thoughts and prayers are with the family members of flight #MH370 . I've asked all measures possible to be taken.
— Mohd Najib Tun Razak (@NajibRazak) March 8, 2014 
Reuters have now reported  that the plane crashed in the South China Sea, however no official announcement has been made at the time of writing.