The balloons are going up at the Royal Academy. Or is the Royal Academy going up in a balloon? As it prepares to open its big autumn exhibition, a great tower of silver baubles is being installed in the courtyard. In the immaculately polished surface of each shining globe, a miniature image of the entire building — grand colonnade, cobbled courtyard and your own gawping figure included — appears to be floating skywards.

This eye-catching sculpture stands in front of Burlington House just as a helium-balloon seller stands at the entrance to a fairground. So roll up, roll up for the great RA spectacular: a show of sculptural work by the perennially popular Anish Kapoor. His retrospective is being hailed as a landmark moment. It is the first time that a contemporary artist will take over the establishment’s galleries in their entirety, for although plans were in progress for Henry Moore to do so back in the Eighties, he died before the project could come to fruition. And it is certainly the first time that an artist has been allowed to install a piece such as Svayambh in the gallery. This massive installation occupies five galleries. A track is laid through them, along which shunts a vast mass of red wax, the size of a train carriage. It scrapes against the sides of the doorways as it passes, leaving behind sticky accretions of wax like mucous.

That Kapoor has been offered so much space is probably as much a response to the sheer scale of his productions (his works are often vast, as anyone who saw his mammoth Marsyas tethered like some stranded zeppelin in the Tate Turbine Hall will know) as to the way that he is viewed within the art world. But comparisons with Moore, the master of monumental bronzes, are nonetheless apposite.

The Mumbai-born Kapoor, who first moved to London in 1973, emerging from Chelsea College of Art and Design in the vanguard of a group of pioneering British sculptors (Richard Wentworth, Tony Cragg and Richard Deacon among them), now fills Moore’s erstwhile role as the creator of public art. Once, no self-respecting city seems to have felt complete without a curvaceous Moore figure lounging about in its square. Nowadays a Kapoor work is seen as a must-have municipal trophy. His Cloud Gate in Chicago, an enormous ellipse of polished steel, proved so popular that it was even awarded its own public holiday — an irony, perhaps, since it cost well over $20 million (£12 million) and citizens should probably have been working overtime to pay for it. And Kapoor is currently working on a five-part series of sculptures that, destined to occupy five northeastern towns, are created on a scale that make his Marsyas look like a maquette.

Now this new Royal Academy exhibition offers a survey of Kapoor’s career through a representative assortment of some two dozen pieces, of which almost a quarter are new. The exhibition opens with a selection of the sort of brilliantly coloured powdered pigment pieces with which Kapoor first came to attention in the Seventies. Progressing via various concavities and protuberances hollowed out of walls or swelling into empty spaces, it leads the spectator past vast enigmatic sculptures and through a hall of mirrored works that set the eyes squinting to such recent pieces as Shooting into the Corner. This 2008-09 installation, powered by an ear-stunning pneumatic compressor, flings big cylindrical bullets of dark crimson gunk right across a gallery where they explode like a body in a bad Hollywood movie: a mass of waxy giblets splattered apart on white walls.

This show is a sure-fire hit. It is bound to prove popular. Who would not respond to Kapoor’s skills as a colourist? Here is an artist who revels in the dramas of scarlet, who can bathe us in the glow of a golden buttercup. He understands our childish delight in sensations of disorientation and so leads his spectators through a hall of mirrors in which gleaming surfaces of silver and gold stretch and distort and flip reflections over, turning you and the rooms around you upside down. It feels as if the whole world has been loaded into a washing machine. You watch through the mirrored window as it tumbles and jumbles about.

Kapoor plays with the intrigue of art. A mammoth rusty form occupies the octagonal gallery — a vast lump of enigma. It looks a bit like a stopper that might fit into the aperture of the glass oculus above. The longer you examine it, the less recognisable it feels — but we are drawn to its mystery as surely as we long to lean into the dark aperture on one side of it and make that “hoo” sound that you call into a cave.

Kapoor’s sculptures are striking, audacious, seductive, spectacular. They stir feelings of awe, sensations of wonder. What do they mean? Screeds have been written. They range from simple description through baffling psychobabble to cosmic conundrums. But forget the waffle. Just look. Here is an artist who plays games with perceptions. Can emptiness become an object? Can the monumental be immaterial? Can the ethereal take solid form? At what point does something come into being and at what point does it vanish back into complete non-existence?

The work of Kapoor makes such musings a sensual experience. Look at his Slug — a vast, shiny red-lipped ovoid that looks as seductive as one of Georgia O’Keefe’s huge sexual flowers is attached to a coil of unspooling intestine still marked with the pencil notes of the technician. The inner and the outer, the made and the unmade, are contrasted in a piece that attracts the spectator just as it stirs a queasy repulsion.

And how can one not respond at a simple physical level to the centrepiece of the show: Svayambh. The Sanskrit title of this piece can be translated as “self-forming” apparently. Kapoor is alluding to the minimalist practices of erasing the processes of making. He is showing us an art work in the process of making itself. We stare in amazement as this incredible object travels slowly back and forth — and perhaps not least because it makes such a mess of Grade I listed galleries.

The spectacle is amazing. But it stuns the mind. And maybe that’s the problem with Kapoor’s work. The spectator stands and marvels at these works as he might admire a magician’s trick. He watches it happen. He may even be invited to come on stage and participate. But he does not have to contribute anything more than his presence. His mind and imagination have not really been engaged.

And yes, the end product is satisfying. But it is precisely such satisfaction that deadens the desire to see farther, to probe more deeply, that the art work should stir. Kapoor translates our most complicated philosophies into a gallery-goer’s version of cinematic special effects. We may admire the technical skill, marvel at the facility, enjoy the experience, but in the long run the profundities are all there on the surface. We are watching optical performances. It is pleasurable and popular and sometimes sensational: but then so is a day at the fairground.

Anish Kapoor is at the Royal Academy of Arts, London W1 (020-7300 8000; www.royalacademy.org.uk), from Sept 26 to Dec 11

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