Appeals are being made in the UK press and social media to find lost loved ones from many nationalities.
The following are some of the people identified as missing:
Acclaimed artist
Photographer Khadija Saye, 24, recently had her work shown at the Venice Biennale.
She was believed to have been living in the tower with her mother, Mary Mendy, who is also missing.
Saye has known Labour MP David Lammy for a "number of years," and the politician led the appeals on Twitter, calling her a "beautiful soul and emerging artist".
'Very, very scared'
Jessica Urbano, 12, has not been located since talking to her aunt Sandra Ruiz at around 1.40am on the day of the fire.
"She was with a group of people in the fire escape, in the fire stairs, they live on the 20th floor," said Ruiz, while appealing for information.
"She would have been in her bed clothes and she will be very, very, very scared. We're just desperate to find her."
Six-month old missing
Farah Hamdan, her husband Omar Belkadi and their six-month-old daughter are still unaccounted for, according to Farah's father, Rkia.
"We've been to all the hospitals and we've been searching all day but we still haven't found them, we just want to know they are safe," he told 'The Sun' newspaper.
Her other two daughters, aged six and 10, were eventually found in hospital.
Eritrean national
Mo Tuccu, an Eritrean working as a security guard for London PR agency Red Consultancy, is missing together with his wife, Amalahmedin and their three-year-old daughter.
A colleague at the consultancy firm told the BBC he was due to win an award for 10 years service to the company.
Italian architects
Italian couple Marco Gottardi, 27, and Gloria Trevisan, 27, are both missing.
The two architects had moved to London three months ago for work.
Marco Gottardi and his girlfriend Gloria Trevisan Source: Facebook/Pamela Pizziolo
"At the moment there are two of our citizens who are confirmed as missing," an Italian foreign ministry official told AFP.
An Italian newspaper, 'Il Mattino di Padova', said Gottardi had been on phone with his father and told him the smoke was rising before the line suddenly cut.
Afghan family
Mohamed Neda's family is believed to have fled the blaze, but he remains unaccounted for.
Friend Ariana Neumann Rodger posted on Facebook that 22-year-old son, Farhad, was being treated in hospital while his mother was in intensive care at Kings College Hospital.
They lived on the top floor and Mohamed 'Saber' was last seen on the stairwell injured and helping neighbours.
Oldest missing person
Sheila Smith, 84, is the oldest of those to be declared missing so far.
Her son Adam, who describes himself as a tech entrepreneur, launched an appeal on Twitter, writing "my Mum is missing -- Sheila, 84 years old, 16th floor Grenfell Tower."
Desperate appeal
Rania Ibrham, 30, uploaded a distressing Facebook Live video as she pleaded for help from a smoke-filled corridor before going back into her flat.
She has not been contactable since.
Bodies inside
The bodies of many of the victims are thought to be still inside the smouldering tower.
Authorities are still struggling to find out how many people were actually in the 24-storey building when the fire started in the early hours of Wednesday.
Up to six hundred people are believed to be residents of the Grenfell Tower.
A "good half" of the building has yet to be searched according to Fire Commissioner Dany Cotton, who explained that additional shoring up was needed for firefighters to access the upper floors.
But Metropolitan Police Commander Stuart Cundy sought to downplay the number of missing people saying that while 400 people had been reported missing, "it is an assumption."
"A lot of people managed to get outside themselves and 65 were rescued," he said.
"From a personal perspective, I like to hope it is not going to be triple figures."
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