Images show the scale of the fire spreading to all floors
Firefighters are still tackling the blaze at Grenfell Tower in north Kensington, where hundreds of people are thought to live.
Eyewitnesses reported seeing people trapped inside their homes.
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.
The Met Police has set up an emergency number on 0800 0961 233 for anyone concerned about friends or family.
Eyewitness: 'We've seen people blinking lights within the building'
At 06:15 BST, London Ambulance Service said 20 ambulance crews had been at the scene, 30 people had been taken to hospital, but the nature and level of their injuries was not yet clear.
Mayor of London Sadiq Khan said a "major incident" had been declared.
Three hours after the fire started, police said people were continuing to be evacuated from the tower.
David Benjamin says he was woken by a neighbour banging on the door
Paul Munakr, who lives on the seventh floor, managed to escape.
"As I was going down the stairs, there were firefighters, truly amazing firefighters that were actually going upstairs, to the fire, trying to get as many people out the building as possible," he told.
He said he was alerted to the fire not by fire alarms but by people on the street below, shouting "don't jump, don't jump".
"Now, honestly I don't know for certain if people jumped off the building to get away from the fire, but the main thing for me with this incident is the fact that the fire alarms didn't go off in the building," he said.
"The police keep pushing back their cordons, pushing back members of the public for fear the building might collapse."
London fire crews said it was a "large and very serious incident"
There are 120 flats in the residential block
London Fire Brigade assistant commissioner Dan Daly said firefighters were "working extremely hard in very difficult conditions to tackle this fire".
"This is a large and very serious incident and we have deployed numerous resources and specialist appliances."
London Ambulance Service medics specially trained in life-saving medical care in hazardous environments have also been sent to the fire.
The first reports of fire in the tower, in Latimer Road, on the Lancaster West Estate, came in at 00:54 BST.
The A40 Westway was closed in both directions. London Underground said the Hammersmith and City and Circle lines have been closed between Edgware Road and Hammersmith.
Smoke could be seen from miles away
Most of the building was alight
According to Kensington and Chelsea Council, the tower block contains 120 flats and is 24 storeys high.
It is managed by the Kensington and Chelsea Tenant Management Organisation (KCTMO) on behalf of the council and had undergone a two-year, £10m refurbishment that was completed last year.
The work included new exterior cladding and a communal heating system.
The local Grenfell Action Group had claimed, before and during the refurbishment, the block constituted a fire risk and residents had warned that access to the site for emergency vehicles was "severely restricted".
The BBC has been unable to contact the property's management company in the hours since the fire.
One eyewitness, George Clarke, the presenter of Channel 4 TV programme Amazing Spaces, told Radio 5 Live: "I'm getting covered in ash, that's how bad it is.
"I'm 100 metres away and I'm absolutely covered in ash.
"It's so heartbreaking, I've seen someone flashing their torches at the top level and they obviously can't get out."
Jody Martin said he ran towards the building to try and help when he saw the fire.
He said he was shouting at people to "get out, get out" but that residents were shouting back that they were stuck as corridors inside the building were filled with smoke.
Tim Downie, another eyewitness, told the BBC part of the building was "completely burned away".
"It has burned through to its very core," he said.
"It looks very bad, very very bad. I've never seen anything like this. It's just such a big fire.
"The whole building is just crumbling. It's just billowing black smoke."
Safiyah, who is about 500m away from the building, said: "There are lots of people gathered in the street. I just see more and more flames burning and tragically I hear people crying for help.
"The entire building is burning through."