And backstage on a drizzly Brussels evening these performers and dozens more from other countries sit together and alone in a spacious temporary dining area, eating noodles, stews and curries, listening to music on headphones, playing cards, drifting in and outside, chatting, yawning, skipping. In their tracksuit bottoms, jeans, puffa jackets or fleeces they seem disarmingly normal.

The scene's beguiling nature is shaped by what preceded it: just half an hour before these average Joes had been twisting their bodies into superhuman shapes, making impossible leaps, juggling mind-bogglingly numerous items, balancing with improbable grace and spinning traditional children's toys to awe-inspiring effect.

And their costumes: the tracksuit look swapped for kohl-eyed make-up, for skin-tight suits dripping with tassels, for shocked-haired wigs and for blue bowlers with a baffling multitude of uses.

"Mesdames et monsieurs. Bonsoir. Welcome to Cirque du Soleil," greets John, the 'Quidam' show's primary character at the top of the show. He's a goofball ringleader with a Tintin lick of hair and spangly, wide-shouldered jacket, far removed from the red frock-coated gent calling traditional circus.

John's otherness sets the tone for two hours spent within an enticing fantasy land. The lights go down and spotlights reveal a living room scene: dad in his armchair with the newspaper, mum and her retro radio and a little girl in yellow between; ignored and bored.

Then a mysterious call and a huge headless shape appears at the door, somehow sporting a bowler hat which drops before the girl. He leaves, mum and dad are swept to the ceiling in their chairs and two virtuosic hours of circus magic ensue.

"It's like Alice in Wonderland," says Roland Richard, Quidam's effervescent general stage manager and the man who draws together the dozens of performers, live musicians and incredible light show to create this breathtaking two hour spectacle.

"I'm the pilot of an amazing aircraft. I decide how to call the show, the music and how to fly people. Quidam brings you to your childhood and the beautiful moments of your life."

Richard, an opera singer before Cirque du Soleil, was with the very first productions of Quidam 15 or so years ago, soon after its creation by the Belgian director Franco Dragone. He has clearly lost none of his passion for the show.

"I've been calling this show for 3,361 and it had done 1,000 shows before this in America. The older shows rely less on machinery than human talent."

That talent reveals itself again and again. There's the German Wheel: a giant hamster wheel manned by a bowler-hatted figure in a skin-tight be-tasselled suit. He spins the wheel from its centre, whirling round and round the stage to a pulsing beat.

And the aerial contortionist in silk, a woman who climbs high above the stage up a long curtain of fabric, draping it round her body, dropping and clinging, teasing her body into elegant shapes.

The diabolo artists offer a prodigious display with these spinning childhood toys on a rope. Three Chinese girls in mini-skirts and on their heads what seem to be silver-painted household funnels. They skip, prance and even back flip across the stage, all the while spinning their diablos, hurling them high into the air for each other to catch and carry on.

The banquine is a real spectacle. Two girls and a group of guys, some short, some tall, but all muscular and primed like springs. They arrange themselves into humanly impossible towers, pyramids and a myriad other shapes, sometimes leaping and flipping into position, other times using their strength and balance to move with languid grace.

I had the privilege of first watching them backstage, practising and honing moves. Sometimes they didn't seem to get it exactly right and a performer would tumble via a safety harness to the mat. I wondered if they would really pull it off on stage. There was no reason to doubt; the banquine is a skilful and stirring example of teamwork in action. The performance climaxes with a move made all the more incredible for its inevitability. The performers form two neighbouring human towers, one of two people, the other of three. You can see Dmitro Sidorenko atop the first and know he's going to leap from shoulders on to the higher tower – but wonder until the moment he pulls off the backward spring if he'll miss his target.

"Before Cirque du Soleil I was in the Ukraine national team for Acro Sports, doing the Euro cups and world cups and one of the circus people came and asked me to join. I pretty much said yes straight away," says the 30-year-old backstage, explaining banquine and how he got involved.

"It's two people holding hands then throwing us in the air and then catching, or doing hand balancing on someone's head or leg or arm and also jumping and landing in pyramids, stuff like that."

He makes his extraordinary performance sound so everyday, but to him it is. Dmitro's been with Quidam since 1995. He met his wife Alicia on the show when she did a hand-balancing act and they now have a two-and-a-half year old daughter, Marina.

From humble beginnings, Cirque du Soleil is now a global phenomenon. Several touring shows criss-crossing the continents, but with origins in street performance in the French-speaking Canadian city of Quebec.

"A group of street performers were doing a street festival in Canada," explains Roland.

"They got together with the founder Guy Laliberte and thought they could present it somewhere else, somewhere bigger and built a show.

"It was voted one of the best shows in Canada and they started getting abroad and now it's got 10 touring shows, five or six in Vegas in Tokyo, Macau, and Dubai – it's huge. It's a circus without animals – it's a joke, but we are our own animals."

At Cirque du Soleil's core, says its manifesto, is a constant need for reinvention. Its diverse range of shows, each with live music, include Zumanity, billed as an erotic cabaret for adults; KA, an 'epic show employing the visual language for film'; LOVE, inspired by the music of The Beatles, and DELERIUM, a musical for arenas. Quidam hits Sheffield Arena this month, having already been seen by over eight million people on four continents. For Roland its appeal lies in mixing awesome circus skills with surrealism and healthy shot of accessibility.

"You'll see images you can identify with Magritte, the painter: the headless guy with the hat, the ladies whose faces are covered with sheer cloth, the umbrella and clouds," he enthuses.

For Roland Cirque du Soleil is about welcoming audiences into a world of escape, of enchantment and wonder:

"When you create magic you don't want to know exactly how you create magic," he says, "you want magic to speak to you."

Cirque Du Soleil is at Sheffield Arena for seven performances from April 9 to 12. For more information go to www.sheffieldarena.co.uk