Either its name comes from the Roman temple of Mons Jovis (Moutain of Jupiter) or from the medieval Jewish cemetery in the south side of the mountain, Montjuïc is an obligatory visit while in Barcelona.
Spite of its more ancestral occupations, it was the World Exposition of 1929 and the Olympic games of 1992 that bring Montjuïc in to the city. The Palau Nacional, other palaces and infrastructures in Montjuïc, including the wide mountain parc, were built for the World Exposition, but the full development of the hill come with the Olympic Games with the construction of the stadium and of several sport facilities.
Just above the commercial port one can find the 17th Century fortress, the Castell de Montjuïc, which today houses the military historical museum. In the days of Franco, the fortress on the Montjuïc was a prison for political prisoners.Today it can be visited and enjoy some great views of the city.
One way to get to Montjuïc is from Plaça Espanya (you have a subway station there). From the square you are drawn by the imposing view of the Palau Nacional with its fountains and cascades. You can get to the top by the stairs or the escalators and visit the Museum of Nacional Art from Catalunya.
At the base of the cascades there is the Magical Fountain built for the World Exposition of 1929. After dark takes place a show of music and light much appreciated by the visitors. At the same level, on the right, you can find the famous Mies van der Rohe Pavillion also built for the 1929 exposition.
Walking through beautiful gardens you arrive at the Fundació Joan Miró, were you can find many of the works of the Catalonian painter. Not far from the museum, across the street, there is the funicular station to go to the castle (alternatively, you can use the bus with is a more budget friendly alternative).
After visiting the castle, while descending the hill (either by bus or by foot), you can visit the olympic stadium area and enjoy both the place as well the great views of the city.