A prayer service was held this morning at nearby St George's Tron Church and flowers were being left close to the crash site. The square was cordoned off as the investigation got under way.
The BBC reported that Police Scotland had asked for anyone with photographs, videos or any type of mobile footage of the incident to contact them by email.
Experts will examine is how the vehicle travelled around 300m in an almost straight line before it came to rest. Eyewitness said they saw the truck knocking people down like "pinballs", including a baby in a buggy. A shopkeeper described how a mother fainted after the truck just missed her and her child.
Police declined to comment on the ages of the victims but said the area was packed with families.
Bystanders said the driver appeared to have had a fit or a heart attack at the wheel.
Emergency services said they feared the number of deaths could rise.
British Prime Minister David Cameron said he was being briefed on the tragedy, saying, "My thoughts are with the families of those involved and the emergency services."
Shopper Melanie Greig said: "The lorry just lost control, it just went along the pavement knocking people down like pinballs. There was a baby in a buggy, there were two little kids. It was quite horrific. People were trying to run out the way, but when something like that comes up behind them how can they? There was noise and bangs and screams."
Findlay Mair, 23, said: "The screaming just kept going on. Suddenly it was just chaos. Some people were running towards the scene, others were just standing stock still, in shock. The people who were hit had no time to take evasive action. They had no chance. I saw one girl who had been hit, lying on the ground. She got up, and blood was pouring from her mouth, then she went down again.
"She was young, of student age. She just screamed and screamed, and then fell down again. She had terrible injuries.
"A woman motorist got out of her car and just stared at what was unfolding and then started sobbing. She was just yelling, 'Oh God, oh God'.
"There was Christmas shopping littered everywhere, among the bodies and bits of the vehicle. And there was blood everywhere. Time seemed to stand still. It was a terrible, terrible scene.
"It was difficult to take in. One minute the square was a typical Christmas scene, with people enjoying their Christmas shopping and the next moment it was just carnage.
"A lot of people were standing there, trying to understand what was happening. Then everyone was calling emergency services."
Police said it did not appear to be a deliberate act and confirmed the driver was among those taken to hospital.
Student Lewis Irvine, 17, said: "I could see the driver at the wheel. It looked like he was having a fit or something. There were bodies lying on the ground. It was awful."
Another witness, Janey Godley, said it was "like something out of a horror movie", adding: "The traffic poles in Queen St were bent like straws as they had been run over."
Reece Miller, 16, a student from Motherwell who was in Glasgow for Christmas shopping, said: "We were having lunch on the corner of the square when we heard a loud bang. The building we were in was shaking. I looked through the window. We saw that a lorry had crashed into the side of the building. I think it hit a car and then a number of shoppers. We went out and saw lots of bodies lying on the floor. Most of them were covered by blankets.
Joty Chopra, a 21-year-old student, said: "We were having lunch around the corner and heard a loud bang and came running out.
"When we got to the scene there were three people lying on the floor - two of them were in a pool of blood."
The crash and the carnage it caused were witnessed by hundreds of members of the public on one of the busiest shopping days of the year.
There is a "Glasgow Loves Christmas" fair in the square, which is popular with families, and restaurants and shops surround the area.
Members of the public ran to the aid of the injured and some tried to give mouth-to-mouth resuscitation before the emergency services arrived.
The accident happened just over a year after 10 people died in the city centre when a police helicopter crashed into the Clutha Vaults bar.
Superintendent Stewart Carle of Police Scotland said he wanted to reassure people it was a road traffic accident and "nothing more sinister". Telegraph Group Ltd
