Melenchon — who is standing under the banner "Unsubmissive France" and who has been an MEP for south-west France since 2009 — was seen as an outsider several months ago, but has seen a surge in popularity, largely due to his huge YouTube following and his use of holograms to attend rallies.
In recent polls, however, he has risen from fifth place to third, behind Le Pen and Emmanuel Macron, who was economy minister under Francois Hollande before resigning to start his own party, called En Marche, which is a pro-EU centrist party. Macron has never stood for election before.
The other frontrunner is Francois Fillon, the former prime minister, who is standing for the Republicans.
He has co me in for some criticism over payments made to his wife and children for work which was allegedly "fictional," for which he is under investigation.
The fact that there are four frontrunners — of which two are extremes, calling for a referendum on French membership of the European Union — means that the two-part vote, April 23 and May 7 could be close.
It is the first time in the Fifth Republic that the two main parties — the Republicans and the Socialists — have faced real competition in what has become a four-horse race.
The latest Ifop-Fiducial poll, April 12, showed Le Pen winning 23.5 percent in the first round, one point ahead of centrist Emmanuel Macron, with Fillon on 19 percent and Melenchon on 18.5 percent.
If Le Pen and Macron lead the first round, then polls suggest Macron would win the second round. However, if there are significant votes for both Melenchon and Le Pen — which is likely — it could lead to calls for a referendum on French membership of the EU, despite them not winning the presidency.
The French have recently been happy to bring about electoral upsets, with a president and two prime ministers being thrown out of the presidency run in the primaries: Nicolas Sarkozy, Alain Juppe and Manuel Valls.
Significantly, another Ifop poll, ranking 30 French politicians in terms of which of the voters had an excellent, good, bad or very bad opinion, shows Melenchon in top position — particularly among young voters — with Macron third, Le Pen 27th and Fillon trailing in 37th place.