Independent centrist Emmanuel Macron will face far-right Marine Le Pen in a run-off on 7 May
Mr Macron, a 39-year-old who has never fought an election before, tells jubilant supporters: "In one year, we have changed the face of French politics"
Marine Le Pen called on "all patriots" to join her and "concentrate on what is essential - the survival of France"
The beaten candidates from mainstream parties, Socialist Benoît Hamon and conservative François Fillon, declare support for Mr Macron
Image source, ReutersPeople wait to vote at Vaulx-en-Velin near Lyon
Turnout at 10:00 GMT was slightly higher than at the same stage in 2012, the interior ministry said.
Four hours after polling stations opened, turnout was just over 28.5%, which would put the final turnout on course to beat the figure of 79.48% in the first round in 2012, AFP news agency reports.
Image source, EPAPresidential candidate Emmanuel Macron of the En Marche movement has cast his vote in the northern town of Le Touquet.
He was accompanied by his wife Brigitte Trogneux, who was his secondary school teacher.
Read more: The meteoric rise of France's youngest presidential candidate
Information for French voters at a London tube station
French voters in Wembley, northwest London, were greeted by this information sign upon exiting the nearby tube station.
None of the wise words, inspirational quotes or lines from poems for which such signs have become famous, however.
Citizens of the land of Baudelaire, Rimbaud and Apollinaire were instead given a masterclass in Anglo-Saxon pragmatism and told the polling station was "in front of Asda" - a supermarket.
Image source, AFPFrance-based English-language news website The Local, external has produced a handy guide for anglophones and other foreigners seeking to understand more of the commentary surrounding the election.
It says the Course à l'Elysée - election race - has heated up to the point where the contenders are coude-à-coude - literally elbow to elbow, or neck-and-neck - and the result is dans un mouchoir de poche - which translates as in a pocket handkerchief but means too close to call.
Another useful term is faire barrage, which means co-operation between parties to prevent another party from winning the second round of voting.
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Andrew Walker
World Service economics correspondent
Image source, ReutersThe winner will have to tackle France's high unemployment rate
Whoever eventually walks into the Elysee Palace will face some difficult, even intractable challenges.
The big picture is an economy with a high standard of living and high productivity but some persistent problems.
The issue that stands out is unemployment. Close to three million people who want to work and are looking for a job don't have one. The unemployment rate, external is 10%.
It's not as bad as some eurozone neighbours, but it is above average for the region and far worse than, for example, the Netherlands and Austria, where it is below 6%, and Germany, where it is below 4%. The figure for the UK is below 5%.
You might think that unemployment will decline as the recovery from the eurozone's recession continues. Many forecasters expect it will, but not by all that much.
The International Monetary Fund estimates, external that it will be hard to get unemployment down much below 8.5%.
Image source, EPAFrench President François Hollande arrives at a polling station in Tulle, central France, to cast his vote.
Faced with historically low approval ratings, he is France's first president in the modern era not to seek re-election.
His presidency was overshadowed by a series of jihadist terror attacks from January 2015.
His plans to introduce a new labour law in an attempt to tackle stubbornly high unemployment - originally his top priority - met with violent protests.
Image source, ReutersThe BBC's Kevin Connolly writes:
In Hénin-Beaumont, Marine Le Pen arrived to vote at 11:00 (09:00 GMT).
Dozens of cameras were waiting to record her making the short walk up the driveway of the Jean-Jacques Rousseau school but she was driven past in a large people-carrier with blacked-out windows flanked by bodyguards on foot.
A few minutes later she departed in the same manner.
There were cameras inside to record the candidate casting her ballot but perhaps for security reasons there'll be no pictures of Madame Le Pen outside the polling station today.
Read more: Is France's Front National leader far right?
Umbrellas are out in a rainy German capital as French expatriates wait to vote.
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One French voter in London, who gave his name only as Nicolas, said the queue at the polling station at the French school in Wembley was so long that it took an hour-and-a-half to vote.
An enterprising croissant seller was doing a roaring trade in the car park, he said.
“The atmosphere was friendly – people were bringing their children to show them how the voting process works,” he said.
Nicolas, who was also casting a proxy vote for his sister, said people in the queue weren’t discussing how they would vote.
A long queue at the French school in Wembley, with the stadium visible in the background
There were long queues too in the French community stronghold of South Kensington, where the line stretched several hundred metres, according to reporter Vincent Cohen.
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Image source, AFP/GettyVoting in Guadeloupe and other western hemisphere territories took place on Saturday
Reporting from French territories in the Americas reveals that participation rates have dropped in comparison to the first round of 2012.
Figures from Saturday afternoon - before voting had finished - reported by French media are as follows:
Those figures were slightly down on similar results from 2012.
The last time the abstention rate was high - in 2002, when the abstention rate was 28% - the Front National led by Marine Le Pen's father Jean Marie Le Pen got through to the second round before losing to Jacques Chirac in the run-off.
The abstention rate in 2012 was much lower at 20%.
But some polls, such as this one quoted in the Financial Times, external, have predicted this year's abstention rate could be as high as 37%.
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A minor flurry of excitement near the polling station here in Hénin-Beaumont.
A stretch limousine arrives and five or six protestors emerge - two are topless women in Le Pen face masks with the words "Team Marine" emblazoned on their chests.
They chant "Marine, President" with ironic intent as they are swiftly subdued by the police.
The Voix du Nord newspaper reports that the women are from the feminist activist group Femen, whose founder was given political asylum in France after causing controversy in her native Ukraine.
Femen activists also protested at a Le Pen speech in Paris earlier in the campaign.
Image source, Reuters"Vote, even if it's not for your man," says Carla Bruni
Several big names in French public life have urged French citizens to get out and vote rather than staying at home.
Writing in the Journal du Dimanche, external, public figures including singer Carla Bruni, respected ecologist Nicolas Hulot, IMF chief Christine Lagarde and writer Jean d'Ormesson have all put their names to the campaign.
"If democracy bores you so much, try dictatorship," quips Mr Hulot, before thundering that resignation and disinterest "make me angry."
Ms Bruni - wife of former president Nicolas Sarkozy - has come up with an election song in which she encourages people to vote "even if it's not for your man", a reference to her husband who crashed out in the Les Republicains primary, losing to eventual candidate François Fillon.
France historically has high election participation rates - the turnout in the first round of the 2012 election was about 80%.
By comparison, the last time that number went to the polls in the UK was in 1951 and the turnout for the 2015 UK general election was about 66%.
How much can we read into this display?
Electoral straws in the wind or random acts of minor criminal damage?
Posters of all 11 candidates are displayed outside the main polling station in Hénin-Beaumont, writes the BBC's Kevin Connolly in the town.
Only one candidate has two posters - that's Marine Le Pen.
Only three have been defaced - the independent centre-leftist Emanuel Macron, the conservative François Fillon and perhaps rather surprisingly the Gaullist Nicolas Dupont-Aignan who could not otherwise be said to have played a particularly prominent role in the campaign.
BBC Europe correspondent Kevin Connolly is at the Jean-Jacques Rousseau school in the northern town of Hénin-Beaumont, where the far-right Front National (FN) candidate Marine Le Pen is due to vote later this morning.
A large crowd of journalists has gathered there, he says. Local police have already summoned a van carrying extra crush barriers to extend the temporary fence around the building.
French journalist Pierre Tremblay has posted a video of the assembled media.
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Ms Le Pen's campaign director David Rachline - who is also the mayor of the town of Frejus and an FN senator - has already voted in the town.
Image source, AFP/GettyMr Rachline (L) is Ms Le Pen's campaign director
Image source, ReutersWho will voters at this Lyon polling station choose?
The world is watching to see if France becomes the next major country to challenge liberal values and the political establishment.
Two of the four candidates leading in the polls occupy the political extremes - the far-right Marine Le Pen and hard-left Jean-Luc Mélenchon.
The victory of either would place a question mark over France's continued membership of the EU in its present form. Were it to leave, could the EU survive the exit of one of its founding fathers?
After the Brexit vote in the UK and election of US President Donald Trump, could France be the next nation to deal a blow to politics as usual?
Le Monde has a piece, external casting doubt on whether the exit polls will be able to call the two top vote-winners straight after polling has ended, given how tight the race could be.
It quotes the chief operating officer of polling firm Ipsos, Brice Teinturier, as saying: "We will not take undue risk. If we are unable to decide between the second and third, we will say so."
Image source, ReutersMany French expatriates have already voted, with hundreds of thousands living in the western hemisphere - including about 120,000 French people living in the US - casting their ballots on Saturday.
Expatriates living in Asia began voting earlier today.
Image source, ReutersVoting at the French consulate in New York took place on Saturday
Image source, AFP/GettyExpatriates in Hong Kong have been voting at the French international school
Image source, Getty ImagesThe five main candidates are (L-R) François Fillon, Benoît Hamon, Marine Le Pen, Emmanuel Macron and Jean-Luc Mélenchon
There are four main front-runners and, as none is likely to secure an outright majority, a run-off between the two leading candidates is now expected on 7 May.
For the first time in 15 years, the far-right National Front has a realistic chance of winning under Marine Le Pen. Centrist Emmanuel Macron has been a leading challenger.
The former favourite, centre-right Republican François Fillon, is still in the race despite a difficult campaign. So is far-left candidate Jean-Luc Mélenchon, who enjoyed a late surge in support. Socialist Benoît Hamon appears to face a tougher task.
For the first time in modern French history, the incumbent - Socialist President François Hollande - is not running for a second term because of poor ratings.
BBC reporter Gavin Lee has just seen this couple limbering up before casting their ballots.
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