European Commission President Ursula von der Leyen released a video celebrating the vaccine rollout, calling it "a touching moment of unity" in the battle to protect the bloc's almost 450 million people from the worst public health crisis in a century.
"We know that the pandemic won't just disappear as of today, but the vaccine is the beginning of the victory over the pandemic, the vaccine is a "game changer", said Austrian Chancellor Sebastian Kurz.
"It is the beginning of the end.it was an exciting, historic moment", she said at Rome's Spallanzani hospital.
Britain recorded another 30,501 positive COVID-19 cases and a further 316 deaths on Sunday, bringing the country's total death toll to 70,752. The vaccine rollout will help the bloc project a sense of unity in a complex lifesaving mission after it faced a year of difficulties in negotiating a post-Brexit trade deal with Britain. The new variant, which British authorities said is much more easily transmitted, has caused European countries, the United States and China to put new restrictions on travel for people from Britain.
"The NHS across the United Kingdom is working incredibly hard to scale up the vaccination programme as fast as they can to make sure everyone on the priority list can get their vaccine easily", said Nadhim Zahawi, the minister overseeing vaccine deployment, in a statement.
"We don't want to waste that one day that the vaccine loses shelf life", Karsten Fischer, from the pandemic staff of the Harz district in the German state of Saxony-Anhalt, told broadcaster MDR.
All those getting shots will have to come back in three weeks for a second dose.
"The MHRA will look at this stuff and decide what's the best thing to do to get the most effective vaccination programme in the fastest, safest way possible".
Germany, where the pandemic has cost more than 30,000 lives, will begin with those over 80 and people who take care of vulnerable groups.
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An investigation is underway into reports that ParCare Community Health Network may have fraudulently obtained doses of COVID-19 vaccines and distributed it in violation of state guidelines. Spain will begin to administer the coronavirus vaccine developed by Pfizer and BioNTech on December 27.
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.
Polish Prime Minister Mateusz Morawiecki called it the patriotic duty of Poles to get vaccinated - a message directed at a society where there is a high degree of vaccine hesitancy born from a general distrust of authorities.
After the first shots were injected Sunday into the arm of 78-year-old Mauricette, a woman in a long-term care facility near Paris, President Emmanuel Macron appealed to his compatriots: "Let's have trust in our researchers and doctors".
Countries are also showing different strategies in their vaccination targeting, with Italy focusing on health workers, France the elderly and in the Czech Republic the prime minister himself at the front of the queue.
Vaccine scepticism remains high in France, with a poll published Sunday by Le Journal du Dimanche finding a majority of people - 56 percent - do not intend to get vaccinated.
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.
Of the 44 percent of people who are in favour of the vaccine, only 13 percent said they were "certain" to receive it. Regional health chief Enrique Ruiz Escudero said the new strain had arrived when an infected person flew into Madrid's airport.