The final -- and by far biggest -- day of EU elections started Sunday with voting due in 21 member countries, including France and Germany, where support for surging far-right parties is being tested.

Polling stations opened in Greece and a string of other member states along the European Union's eastern edge, with voting elsewhere set to begin shortly.

It is a crucial time for Europe. The continent is confronted with the war in Ukraine, global trade and industrial tensions marked by US-China rivalry, a climate emergency and a West that within months may have to adapt to a new Donald Trump presidency.

The vote outcome will determine the bloc's next parliament and indirectly the makeup of the powerful European Commission -- thus helping to shape European Union policies over the coming five years.

While centrist mainstream parties are projected to hold most of the incoming European Parliament's 720 seats, polls suggest they will be weakened by a stronger far right pushing the bloc towards ultraconservatism.

Many European voters, hammered by a high cost of living and fearing immigrants to be the source of social ills, are increasingly persuaded by their populist messaging.

- Battleground France -

France will be the EU's high-profile battleground for the competing ideologies.

With voting intentions above 30 percent, Marine Le Pen's far-right National Rally (RN) is predicted to handily beat President Emmanuel Macron's liberal Renaissance party, polling at 14-16 percent.

Le Pen is hoping to form a far-right supergroup in the European Parliament. But analysts predict disagreements with other hard-right parties -- especially over military help for Ukraine, something Le Pen is leery of -- will scupper that.

In Germany, Europe's biggest economy, the election could likewise deal a blow to Chancellor Olaf Scholz -- whose centre-left SPD is polling behind the far-right Alternative for Germany (AfD).

Leading the polls are the centre-right Christian Democrats, credited with 30 percent of votes -- but on 14 percent the AfD is either neck-and-neck or ahead of all three parties in the ruling coalition: SPD, Greens and the liberal FDP.

That is despite a hit from mounting scandals linked to its lead candidate Maximilian Krah, who is under investigation for suspected links to Russia and China, and sparked a furore with comments excusing the Nazis' feared SS.

Le Pen, who has strived to shed the RN of its past reputation for anti-Semitism and xenophobia, has made overtures to Italy's far-right premier Giorgia Meloni with an eye to teaming up.

But Meloni, while fiercely opposed to undocumented asylum-seekers entering Europe, has cultivated a pro-EU position and given little heed publicly to Le Pen's offer.

- Von der Leyen's ambition -

Unlike Le Pen, Meloni aligns with the overall EU consensus on maintaining military and financial assistance to Ukraine and encourages its ambition to one day join the bloc.

Meloni is also important to European Commission chief Ursula von der Leyen's bid for a second mandate, which will be decided by EU leaders but also needs majority assent in the new European Parliament.

Von der Leyen, a conservative former German defence minister, has opened the door to her European People's Party (EPP) -- projected to come top in the EU parliament but without a majority -- working with Meloni's far-right lawmakers.

Mainstream leftist parties fear that could trigger a sharp rightward turn -- with tougher immigration rules and a watering down of climate policies to appease angry farmers and focus on boosting industrialisation.

It could also further bring the far right into the mainstream, as has happened in Italy and the Netherlands where they dominate governing coalitions.

- Far-right 'normalised' -

Far-right populism and nationalism are already forces to be reckoned with in Poland and Spain. In Hungary, the ruling Fidesz party of Prime Minister Viktor Orban has been blocking further EU aid to Ukraine.

"The far-right has not only become a stable feature of EU politics -- it has also been normalised and is no longer a fringe phenomenon in the majority of member states," noted political analyst Zsuzsanna Vegh of the German Marshall Fund.

More than 360 million voters were called to vote across the EU over four days, with projected overall results due late Sunday evening.

Polling data compiled by Politico suggest the centre-right EPP will win 173 seats in the legislature, with the centre-left Socialists and Democrats on 143 and Renew Europe -- the centrist grouping including Macron's party -- on 75.

The main far-right grouping, the European Conservatives and Reformists, in which Meloni's Brothers of Italy party sits, was projected to win 76 seats.

The smaller Identity and Democracy grouping that includes Le Pen's RN was predicted to get 67.