From the Royal Palace (Kongelige Slottet), walk east along Karl Johans Gate, Oslo's main promenade. To your right is a large courtyard and three yellow buildings, which were part of the old Universitet -- today they are used only by the law school. Murals painted by Munch decorate the interior walls of these buildings. Around the corner from the university on Universitetsgata is the Nasjonalgalleriet. You can enter to see a few of the hundreds of Norwegian, Scandinavian, and other European works. To the rear of the National Gallery, across a parking lot, is a big cream-brick art nouveau-style building housing the Historisk Museum. Take an hour or so to view the impressive collection of Viking artifacts on display. Afterward, continue along Frederiksgate back to the university courtyard and cross Karl Johans Gate to Studenterlunden Park and the Nationaltheatret. This impressive building is not only the national theater, but a popular meeting place -- many buses stop out front, and the suburban train line and T-bane (short for tunnelbane, which is an underground railway, or subway) is right beside it.
Walk farther down Karl Johans Gate to see Stortinget, the Norwegian Parliament, facing the castle. Then go back to Stortingsgata, the parallel street to Karl Johan on the other side of Stortinget. Head back in the direction of the Nationaltheatret and then turn left on Universitetsgata. Walk just a block to reach the redbrick Rådhuset, its two block towers a familiar landmark. After visiting Rådhuset, end your tour with an øl (beer) at one of the many outdoor cafés at Aker Brygge on the waterfront.
The walk alone should take no more than two hours. If you happen to be at the Royal Palace at midday, you might catch the changing of the guard, which happens every day at 1:30. Note that many museums are closed Monday.
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