The National Museum of Korea is a wonderful tourist destination in Seoul. I recommend it very highly.
The modern museum (built in 2005) is very spacious and filled with wonderful works of art. It provides a sense of the long history of Korea. It is very well laid out and includes enough information on the works to give you a sense of the context.
The museum displays relics and artifacts in six permanent exhibition galleries: Prehistory and Ancient History, Medieval and Early Modern History, Calligraphy and Painting, Asian Art, Sculpture and Crafts as well as one gallery highlighting donations.
Yongsan Family Park surrounds the museum and provides a nice opportunity to walk if the weather is nice (as it was when I visited).
It is the sixth largest museum in the world in terms of floor space with a total of 295,551 square meters.
When I visited there was an orchestra playing in the courtyard, and I captured this video:
Nearby (within the park) is the National Hangeul Museum (covering the Korean alphabet, Hangeul, created in 1446).
The “Gyeongcheonsa Ten-Story Pagoda” (경천사 십층석탑, 敬天寺十層石塔) was originally erected at the monastery Gyeongcheonsa in the fourth year of King Chungmok of Goryeo (1348). In 1907, it was illegally smuggled to Japan. In 1960, it was restored in Gyoengbokgung Palace, but proved difficult to be conserve because of acid rain and weathering so it was moved into the museum.
Related: Temple of Literature (Văn Miếu – Quốc Tử Giám, 文廟) Hanoi, Vietnam – Islamic Arts Museum Malaysia in Kuala Lumpur – Boston Museum of Fine Arts and Boston Museum of Science (USA) – Cheng Ho Cultural Museum, Malacca