Eyewitnesses described people trapped in the burning Grenfell Tower, in north Kensington, screaming for help and yelling for their children to be saved.
Firefighters, who rescued many people, were called at 00:54 BST and are still trying to put out the fire.
Police say there may still be people in the building who are unaccounted for.
The ambulance service said 68 patients had been taken to six hospitals across London, with 18 in critical care. A further 10 patients made their own way to hospital.
During the night, eyewitnesses said they saw lights - thought to be mobile phones or torches - flashing at the top of the block of flats, and trapped residents coming to their windows - some holding children.
It is understood that "several hundred" people would have been in the block when the fire broke out shortly after midnight, most of them sleeping.
