Detail from Vulcan’s Forge, by Adriaen de Vries.

A few steps into this exhibition my heartbeat started to race and my head was pounding. It contains among many woozy pinnacles of art a massive and imposing sculptural masterpiece by none other than Leonardo da Vinci.

Officially Leonardo never finished his only attempt at bronze sculpture, a colossal horse he tried to cast in Milan. Officially the work here, a group of three towering figures that usually stands high on the baptistery in Florence, Italy, is by his friend Giovan Francesco Rustici. But the 16th century art writer Giorgio Vasari claimed Leonardo collaborated with Rustici on this masterpiece and here, up close, you can see that he did. One of the awe-inspiring figures has the bald head and "nutcracker" profile of a Leonardo da Vinci caricature. The whole group is like one of his sketches cast in metal. He surely shaped this eerie work of genius, with Rustici modestly acting as technician.

Bronze is miraculous. The metal, and the exhibition. Bronze was the first alloy to be discovered and used by prehistoric people: mixing copper and tin, two soft metals, gives resilient bronze. The artistic potential of this resolute metal is exploited to the full by the statue that opens the show, an ancient Greek sculpture of a dancing satyr. The sensitivity and fluidity of bronze allows this dancer to flex and exult with infectious abandon. Ancient Greek artists created most of their masterpieces in bronze yet almost all are lost, remembered through marble copies. This is a dazzling survivor, dredged from the sea off Sicily.

From here on it's a rush of treasures. A daunting Etruscan sculpture of an ancient monster, the Chimaera, that has been exhibited in Florence since it was found in the Renaissance, is one of the many amazing loans to this show. Italy has opened up its museums almost drunkenly to the Royal Academy. The famous icon of Florence, a stupendous life-size bronze wild boar affectionately known as Il Porcellino, is here.

Il Porcellino snorts its authority in a whole room dedicated to bronze sculptures of animals. This exhibition disdains chronology and geography to display pieces from all times and places side by side. In the room with the Porcellino, a Louise Bourgeois spider crawls up the wall and a Picasso baboon squats pugnaciously.

This bestial Picasso is an explosive work of genius that fires mutually off the spiky power of the Renaissance artist Giambologna's nearby statue of a cocksure turkey. I always thought Giambologna's animals looked like Picassos and this show proves it by putting them side by side. Brilliant.

Elsewhere, two beer cans cast in bronze and painted by Jasper Johns sit majestically in a vitrine. Here surely is a kind of modern art with no historical precedent. Johns made these Ballantine Ale cans in 1960 after fellow New York artist Willem de Kooning joked that Johns's dealer, Leo Castelli, could sell anything, even a beer can.

Today's art, right? Utterly modern, right? But in another case you come across a deceptive replica of a tree, cast in bronze by an ancient Roman artist to serve as a candelabra. The Roman candelabra and American beer cans share a profound love of everyday life and artistic trickery.

If the museums of Florence have been generous so has the National Museum in Lagos. Two faces cast in Ife, west Africa, in the 14th to 15th centuries stand out in the exhibition's final room, an array of metal heads. Their compassionate, precise rendering of human features, unsmiling and serious, stopped me in my tracks. The artist communicates a deep consciousness of the gravity of art – these faces seem to know they will be looked at in hundreds of years in a museum.

Their sense of occasion is appropriate. One of the oldest works in the exhibition is a bronze sculpture of the chariot of the sun, made in northern Europe in the 14th century BC and found in Trundholm, Zealand. The newest, made this year by Anish Kapoor, is a shining circular metal mirror that also resembles the sun. From sun to sun, fire to fire: in one of the last rooms Shiva dances, surrounded by flames as if remembering the casting process.

All human life is here, cast in bronze. It is a moving and inspiring encounter with some of the greatest art in the world and you can't ask more of a blockbuster than that.