The Palacio de Madrid, aka the Royal Palace, is one of the most popular tourist sights in the city. The Palace is the the official residence of the King of Spain. There are many interesting sights around the palace, and we particularly enjoyed strolling around the west side gardens and hanging out in the Oriente Square. The picture below is the east facade of the Royal Palace seen from Oriente Square.
Below the palace are the gardens of the Campo del Moro (Field of the Moor, named because a Moorish army camped here in the 12th century). The gardens were made into a public park in the 19th century, and includes interesting plants, fountains and curving paths.