Described by the maestro himself as his best film, "Goodbye to Language" is a frenetic patchwork of vivid scenes and philosophical musings interspersed with stirring music, brutal edits and voices talking over other voices.
The 83-year-old himself, a resident of Rolle in the canton of Vaud, was noticeably absent from the premiere of his film, sending the cast in his place, but he provided festival organizers with a typically cryptic video message.
"Being elsewhere than here, it is not possible for me to be with you, dear comrades, on May 21st, in fact it is no longer a film even though it is my best," he said.
'The idea is simple'
The Guardian's Peter Bradshaw described the plot as "incomprehensible", The Hollywood Reporter understood it as an exploration of the dynamics in a couple's relationship.
But for the official description of the film in the festival booklet, "the idea is simple".
"A married woman and a single man meet. They love, they argue, fists fly," it says.
"A dog strays between town and country. The seasons pass. The man and woman meet again. The dog finds itself between them.
"The other is in one, the one is in the other and they are three."
One scene sees a naked couple talking about having a baby -- the man wants one but the woman would rather have a dog.
Another sees a man defecating on a toilet while talking to the woman about equality.
Shots of a bath filling with blood, autumn leaves and old black-and-white films recur, while gun shots are occasionally heard in the background.
"Is society ready to accept murder to scale back unemployment?" one woman asks.
Godard turned European filmmaking upside-down in the 1960s with his "New Wave" cinema that shunned studio sets in favour of outdoor shoots, improvised scripts and natural sound.
He has never won an award at Cannes where his films have sometimes been panned. This year his film, "Goodbye to Language" is one of 18 competing for the top Palme d'Or prize.
Critics described the film experience as "curious and funny", "mad, choppy", "breathtaking", or "chaotic, eccentric, exasperating and mad".
They seemed particularly taken with the performance of Roxy the dog, who makes frequent appearances in the film barking, playing in snow, lying on a sofa or whining.
"Loved Godard's Goodbye to Language -- the opium dream of a dyslexic dog. Other interpretations are available," Kate Muir, chief critic of the London Times, tweeted.
Like others, Muir suggested the dog should take the unofficial Palme Dog award, handed out each year for the best canine performance.
'Most interesting use of 3D ever'
In an interview with France Inter radio earlier Wednesday, Godard also saluted Roxy, saying his osteopath had told him that dogs do not communicate, they "commune".
"And with the experience I have with Roxy, it's completely true," he said.
The original use of 3D in the film, while sometimes slightly nausea-inducing, was also praised.
"Godard's Goodbye to Language is his best in decades. Most interesting use of 3D ever. Different shots in either eye!" tweeted The Irish Times film correspondent Donald Clarke.
But while most viewers were left scratching their heads in bewilderment, the mere fact that a film by the legendary Godard had been made was enough to satiate their appetites.
"Finding out about a new Godard movie is like discovering that Che Guevara survived the CIA assassination attempt in the Bolivian jungle, and has just pulled off another bank robbery in some La Paz suburb, raising cash for the imminent revolution," Bradshaw wrote.