Full marks to the National Gallery. In their first exhibition dedicated to Pablo Picasso they have given us something rather different. In Picasso: Challenging the Past they draw on their own and Parisian collections to pitch the greatest artist of the 20th century against the great European painting tradition. They take rarely-regarded aspects of Picasso and piece them together into a new picture of the artist, rather like he himself did inside his head to create his own audacious paintings.

Picasso had a voracious appetite for imagery, the Old Masters seen in museums, the Prado, the Louvre, and from childhood on the art seen in his father’s studio. Moreover, he could hold on to all he saw in his astonishing visual memory — and later on, at will, he would piece them together into ‘something else entirely’.

These transformations, borrowings from and homage to the art of the past — ‘quotations’ he called them — sometimes bold, sometimes allusive, sometimes parodic, are what we now see at the National Gallery.

It’s a very accessible exhibition. They have gathered more than 60 of his seminal works and arranged them according to the great themes of European art history and Picasso’s own career: self portraits, nudes, male and female character types, still lifes, models and muses, and the artist’s later ‘Variations’.

They start with self portrait, Picasso himself always being one of the great themes of his art and life, and also an enduring theme of many of the artists he most admired.

There are eight here painted at various times showing different guises, styles and degrees of complexity, from the 16-year-old’s Goya-esque dandy Self Portrait with a Wig (1897), to a Minotaur alter-ego, to the famous 1906 portrait of Picasso the artist wearing an open-necked shirt and with palette in hand.

This relatively simple image, its sculptural quality deliberately recalling Cézanne, the mask-like face possibly Gauguin, shows Picasso beginning to break away from traditions of representation.

Here as well is a virtual Van Gogh, Man with a Straw Hat and an Ice Cream Cone, painted in 1938, the year after Guernica, the painting that achieved him worldwide fame. In it he gives himself a tattered sun hat, probably a studio prop, and a prickly red, green and black beard, to pay homage to the Dutch artist.

He also pokes fun at tradition. Chris Riopelle, exhibition co-curator, says Picasso often used “a touch of comedy”. He added: “Even in the midst of all that angst Van Gogh must have enjoyed time-out for things like ice cream.”

Moving on via a gallery devoted to the female nude that again shows how varied Picasso’s art was, the nudes going from neo-classical to semi-surrealist to one in full Cubist mode, from statuesque bathers to women at their toilette or reclining, along the way emulating Ingres, Degas, Cézanne, Manet among others, we arrive at a room given over to masculinity.

Picasso was obsessed with maleness, says Riopelle. He wanted to show what it meant to be a man in this world, a Spaniard above all. Pausing in front of a swashbuckling caricature of a musketeer with his costume and ruff evoking Velázquez, Riopelle says we shouldn’t take Picasso’s work too earnestly, that many here “show how funny he can be”.

Think of Picasso and you think of beautiful women, of wives and mistresses and lovers. So, in room four we go back to women. This time they’re not nude, they’re ‘pensive’. Picasso draws on Toulouse Lautrec, Degas, Manet to create works such as Absinthe Drinker (1901) and At the Moulin Rouge; he uses Ingres and Goya’s influence, translating it into the dark, melancholy Fernande in a Black Mantilla, tears of paint seeming to run down the picture surface.

Further along, there are three from the 1920s of Olga Koklova, his first wife, a dancer with the Ballets Russes. One is the most surprising picture in the show: a neo-classical painting, a dignified Portrait of Olga that you’d never think was a Picasso.

The high point of this fascinating exhibition has to be the final room, however, showing Picasso’s ‘Variations’.

The room illustrates his increasing obsession with art history. Late in life, from 1954 to 1962, aged seventy-plus, he worked on a series of bold ‘variations’ of masterpieces from the 17th and 19th centuries.

They include Velázquez’s Las Meninas, a picture he’d first seen when he was 14 at the Prado in Madrid: three of his versions of the Infanta in her unmistakeable box-like skirts (he did more than 50); Delacroix’s Women of Algiers, the first series he produced, working in a frenzy of activity over two months in 1954 on 15 paintings and many more drawings; Manet’s Le Déjeuner sur l’Herbe, an iconic painting Picasso reduced to a palette of blues and greyish-greens (he did 27, plus hundreds of drawings, and prints and sculptures); and, Picasso’s take on the Classical story of the Sabine Women, working from Poussin and Jacques-Louis David’s versions projected on to his studio walls.

He was never a slavish imitator, the curators insist. He turned his influences into his creations, thereby affirming his roots in the Classical tradition that considers imitation essential to creativity.

Or, maybe as Picasso and the T-shirt in the shop says, “Bad artists copy. Good artists steal”.

On top of the 60 paintings in Picasso: Challenging the Past, there’s a display of Picasso’s prints in room one near the National Gallery’s main entrance.

It is worth finding your way to it. Fifteen prints show how Picasso ‘steals’ from Degas, Cranach, Manet and Rembrandt, and as with the exhibition in the Sainsbury Wing, it’s on until June 7.