REMEMBER LYING in bed as a child, unable to sleep? Remember how in the half-light that leaked in from the landing everything familiar seemed suddenly frightening, how every object seemed sentient and every shadow tensed to spring? This is the sort of feeling that a new show at the Royal Academy now conjures. Aztecs takes the imagination captive. It tempts the viewer into a spellbinding pre-Hispanic realm, luring him ever deeper into the violent intricacies of its culture, making him a prisoner of its darkly savage dreams of coiled snakes and cavorting monkeys, of squatting gods and knives that grin.

The Aztecs rose to prominence in 1325, when a vagrant people who claimed to have originated from a place called Aztlan set off in search of the homeland promised to them by their legends. They knew they had found it when, taking refuge amid the marshy islands of Lake Tetzcoco, they saw a long-foretold sign: an eagle perched in a prickly pear devouring a serpent (the image still emblazons the Mexican flag). And so the city of Tenochtitlan (modern day Mexico City) was founded.

Over the course of two centuries, protected by glittering waters and a circle of mountains, it grew into a wondrous metropolis. “Some of our soldiers asked if it was not a dream,” recorded an arriving conquistador, enchanted by his first glimpse of this canalised city with its pink-tinted stone palaces and central sacred precinct of temples and pyramids. It seemed to the incoming Spaniards the fulfilment of some late medieval romance.

And yet they conquered it. Setting out from Cuba, the direction from which the Aztecs had always expected a bloodthirsty deity to arrive and wreak vengeance, Cortés seized fame by claiming this magnificent capital for the Spanish crown. A civilisation still in its prime was destroyed without mercy, its temples vandalised, its treasures melted or smashed to make rubble for the buildings of an emerging New Spain.

For generations the central walled city, the great Temple Mayor, was assumed to be lost. But then, in 1978, routine maintenance work in the old colonial centre of Mexico City uncovered a carving which pointed archaeologists to the exact location of the sacred precinct that formed the heart of the Aztec civilisation. Recently unearthed discoveries, displayed alongside many of the finest and most famous Aztec pieces borrowed from collections the world over, form the core of the Royal Academy show.

This exhibition is quite simply unmissable. The mysteries of a lost empire are entombed in its treasures. Many of the pieces have never been exhibited outside Mexico. Many of them may never be lent again. It would be madness to pass over this unprecedented opportunity to admire them now (and to buy the catalogue).

Here is a rare wooden statue of a fertility goddess which, surviving the systematic bonfires of idols, was discovered only in the late 19th century, when she was still being worshipped by the people who possessed her. Here is a fearsome Cihuateotl, a woman who died in childbirth and became a malevolent spirit, which, with its threatening claw fists and macabre grimace, was until a decade ago used as a scarecrow in Scotland. Here are the gruesomely decorated blades with which priests ripped open the sternum of their sacrifical subjects and extracted the still beating heart. Here is the head of a youth, his inlaid stone eyes burning red in their sockets as a sign that he has drunk pulque, an hallucinogenic cactus juice. Here is a monumental ceramic eagle-man so delicate that it has never before been trusted to travel.

Here is the sort of rare gold work (most precious metals were melted down by the Spaniards) of which Dürer (whose father was a goldsmith) said: “I have never seen in all my days that which so rejoiced my heart.” And here, even more precious to the Aztecs (they considered gold to be the excrement of gods), are the most exquisite surviving pieces made from sacred feathers.

You don’t need to know much about the Aztecs to thrill to this show. The very strangeness of the pieces makes them almost more mesmerising. They have a visceral impact, a power which operates on some atavistic level.

There are informative texts to talk the amateur through the sections, tracing the progress of Aztec culture from its initial appropriation of the conventions of those it had conquered to its eventual dissolution as the Spanish put its artisty to Christian service.

But what the visitor most needs is a capacity for enchantment. This is a world as weirdly seductive, as entrancingly strange as the unpronounceable names of the deities who dominated it. Stare at each piece, just as, as a wakeful child, you might once have stared at the objects in your bedroom. Slowly a world assembles itself around you, at first too huge and close to focus on, but you can sense it breathing in the stillness, hear the thump of its lungs against the stone.

These artefacts — however gravely poised the human figures, however closely observed the animals, the coyotes and rattlesnakes, the scorpions and toads — are never simply decorative objects. The Aztecs inhabited an animistic realm. They crawled like fleas through the pelt of a living universe. Every part of creation was infused with spirit, every element of the world instilled with symbolic magic. Beneath the dense solidity of once-painted surfaces resided the divine essence of the world.

The more you stare mesmerised, the deeper you sink into the mind of the Aztecs, into a universe built upon the bones of the dead. It is a gruesome realm: a place of nightmarish deities, of hummingbirds which feast on human hearts, of obsidian moths and feathered serpents, of priests who preside over ceremonies in which the blood of the slaughtered cascaded down the steps of the pyramids, who robe themselves in the skins of flayed corpses. It is a world of warriors who, kitted out in plumed finery, wage “flowery wars”, taking enemies captive, using weapons to massacre chained prisoners armed only with feathers.

This is a world of terrible beauty. Death springs to life. To understand this is to unlock the alchemical magic of Aztec myth.

• Aztecs is at the Royal Academy (020-7300 8000) from Saturday until April 11