Museo Nacional Centro de Arte Reina Sofía

Listen for a moment before you go inside. Out here near Atocha, the traffic hums and buses sigh at the lights. Then your eye goes to two glass towers rising up the front of a pale stone building. Inside them, lifts glide up and down. The old facade and the clear glass sit side by side. That pairing of old and new is the best introduction to what waits behind these walls.

This is the Reina Sofía, Spain's national museum of modern and contemporary art. It carries the name of Queen Sofía, who was queen consort for much of the democratic era. The Prado shows the old masters, and the Thyssen-Bornemisza covers the long story of European painting. This museum takes the twentieth century and what came after. Together, the three form a walking route that locals call the Paseo del Arte, the Art Walk.

The building did not begin as a museum. It began as a hospital. In the eighteenth century, under King Charles the Third, Madrid built a large General Hospital on this site. The Italian-born architect Francesco Sabatini worked on the design, and today the main wing still carries his name. Look at the long rows of windows and the heavy, plain walls. They were built for ventilation and order, not for display.

For almost two hundred years, the sick of Madrid came here. The hospital closed in the twentieth century, and for a while its future looked uncertain. Some people wanted to demolish it. Instead, it was protected and restored. In the 1980s, it reopened as an art centre, and soon afterwards it became a national museum. The wards and corridors became galleries. You can still feel that history in the long hallways.

Now look again at those glass towers. They were added during the conversion, so the old building could hold modern lifts without cutting into its historic stairwells. Ride one if you can. As you rise, the rooftops of the Lavapiés neighbourhood spread out below you. Few visitors expect a view in an art museum, but this one gives you a quiet panorama of southern Madrid.

Behind the Sabatini building, the museum grew again in the early 2000s. The French architect Jean Nouvel designed a new extension, which opened in 2005. Its most memorable feature is a huge red roof that floats over an open courtyard. The roof covers galleries, a library, and auditoriums. Stand under it and the light turns warm, almost the colour of terracotta. It feels like a covered town square.

But most people walk these halls for one painting. On the second floor hangs Guernica, by Pablo Picasso. He painted it in 1937, during the Spanish Civil War, after German and Italian aircraft bombed the Basque town of Guernica in support of Franco's forces. The Spanish Republic commissioned the work for its pavilion at the World's Fair in Paris that year. Picasso turned the news into a scream on canvas.

When you reach the room, step back first. The painting is enormous, almost the width of a small house front. Everything is black, white, and grey. There is no colour at all, and that makes it feel like a newspaper photograph. You will see a horse in agony, a bull standing over a woman holding a dead child, a fallen soldier, and a lamp held out into the darkness.

Here is something many visitors miss. For decades, Guernica did not live in Spain. Picasso asked that it stay away until Spain had democratic freedoms again. It spent many years at the Museum of Modern Art in New York. It came to Madrid only in 1981, after the dictatorship ended and Picasso had died. The painting moved into this building in 1992, and it has stayed here since.

Near the painting, look for a series of black-and-white photographs. The photographer Dora Maar, Picasso's partner at the time, recorded the canvas as it changed while he worked. You can follow figures moving, disappearing, and returning. It is a rare chance to watch a masterpiece grow, almost step by step. Many people walk straight past these images, but they show how hard Picasso fought with the composition.

Guernica is the centre, but it is not the whole story. The collection follows Spanish art through the century. You will find Salvador Dalí's strange dreamscapes and Joan Miró's floating signs and stars. There is the Cubist work of Juan Gris, and the bold paintings of Maruja Mallo, a woman of the Madrid avant-garde whose work is now finally getting more attention. Later rooms bring in the textured, earthy canvases of Antoni Tàpies.

The museum likes to show art together with its time. Many rooms place paintings next to films, posters, magazines, and photographs. You may hear the flicker of an old projector or see protest graphics on the wall. The idea is that art does not live in a sealed box. It grows out of politics, war, cities, and ordinary life, and the galleries are arranged to make that link easy to see.

Before you leave, find the inner garden of the old hospital. It is a quiet square of trees and paths, framed by Sabatini's arcades. Sculptures stand among the plants, and the noise of the street fades away. In the Nouvel courtyard, look up at a tall sculpture by Roy Lichtenstein that turns a single painted brushstroke into a solid object. It is a playful note after the weight of Guernica.

As you walk back out toward Atocha, look once more at the glass lifts sliding against the old stone. Behind you, a hospital that once cared for the bodies of Madrid now holds the memory of its century. The city keeps moving at the traffic lights. And somewhere upstairs, in grey and white, a lamp is still held out into the dark.