But polls have shown many Europeans are unwilling to take the vaccine, which could impede its effectiveness in beating the virus, while it will take months for large chunks of the population to be immunised.
In a sign of impatience, some European Union countries began vaccinating on Saturday, a day before the official start, with a 101-year-old woman in a care home becoming the first person in Germany to be inoculated and Hungary and Slovakia also handing out their first shots.
This comes after the European Medicines Agency (EMA) and the European Commission authorised the German-US Pfizer-BioNTech vaccine.
Chief Executive of AstraZeneca, Pascal Soriot, claims that his product is as effective as the Pfizer-BioNTech vaccine, which was the first in Britain and the Western world to be granted government approval. She told regional newspaper Le Progres that she supports vaccination, but is concerned that doctors don't pay enough attention to individual health issues.
"Very good, thanks", Araceli Rosario Hidalgo Sanchez said after getting the vaccine, as quoted by the TVE broadcaster, adding "let's see if we can get the virus to go away". According to French media, there will be around 20 more vaccinations done on Sunday.
But even then, Hungary on Saturday jumped the gun on the official roll-out by administering shots to frontline workers at hospitals in the capital Budapest. Each country is deciding on its own who will get the first shots - but they are all putting the most vulnerable first.
In Germany yesterday, 101-year-old Edith Kwoizalla, who lives in a retirement home, received the first of her two shots. In the Czech Republic, Prime Minister Andrej Babis was among the first people inoculated, as vaccinations began nationwide.
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France's first vaccination wasn't broadcast on live television as it was elsewhere, and no government ministers attended.
Public health officials said on December 24 that more than 600,000 people had received the first of two doses of the Pfizer vaccine.
Noting that France's infections are rising again in some regions, particularly among older people in rural areas, Health Minister Olivier Veran warned in an interview with the Journal du Dimanche newspaper published Sunday that pressure on hospitals could start growing again, and said the government isn't ruling out a third lockdown.
Estonia, on its part, vaccinated Jelena Rozinko, a doctor from the Ida-Viru Central Hospital in the city of Kohtla-Jarve, according to the Postimees news portal.
"We'll look back on the advances made in 2020 and say: "That was a moment when science really did make a leap forward", said Jeremy Farrar, director of the Oxford University Clinical Research Unit, which is backed by the Wellcome Trust. "I'm obviously not saying it doesn't exist because lots of people have died of it, but for now I wouldn't have it (the vaccine)".
Franc Kramberger, the former archbishop of Maribor, became the first person in Slovenia to receive the Pfizer/BioNTech vaccine against COVID-19 on Sunday, the country's government said in a press release. Spain, France and Germany, among others, are vowing to put the elderly and residents in nursing homes first.