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Photo: J.L. De Diego, Public domain via Wikimedia Commons
Don't let the sound of it frighten you. This large, well-designed museum opened again in April 2014 after several years of renovation works. It houses an incredible collection of archaeological finds from across the peninsula. It leaves the visitor with a sense of the chronology of civilization in Spain (Iberian, Phoenician, Greek, Roman, Visigoth, Arab, and into the modern age). The famous Dama de Elche, an Iberian (pre-Roman) goddess statue, is in this museum. There are also a few pieces from ancient Egypt and Mesopotamia.
The National Museum of Archaeology is a archaeology museum in Madrid, Spain. It is located on Calle de Serrano beside the Plaza de Colón, sharing its building with the National Library of Spain. It is one of the National Museums of Spain and it is attached to the Ministry of Culture.
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Juan March Institute
A few blocks off high-end shopping street Calle Serrano, this private foundation houses free temporary exhibitions of (mostly 20th-century) art which are often worth checking out. 10-15 min walk from either Museo Lázaro Galdiano or National Archaeology Museum.
Museum of Lázaro Galdiano
This museum houses the stunning collection of Spanish entrepreneur José Lázaro Galdiano (1862-1947) and is considered to be one of the best private collections in Spain. Not only will you find works by Goya, Velázquez, El Greco and others - the museum is also filled with jewelry, furniture, sculpture and ceramics. It is located in Lázaro Galdiano's rather lavish former residence. This is an excellent museum that is usually not crowded and well worth the price of admission.
Las Ventas / Las Ventas Bullring
Arguably the most important bullfighting site in Spain. Unless you find this spectacle distasteful, this is a must see if you visit Madrid during the bullfighting season (May, during San Isidro). Tickets may nevertheless be expensive and hard to get for the more important corridas. Anyway, it usually is used as a venue for shows and concerts.
One of the most beautiful parks in Madrid. Built in 1797-1839, it has a strong Romanticism influence. Declared as an Historic Garden, its lakes with swans and ducks, labyrinths, palaces, squares and fountains makes this a lovely place.