1. Amalienborg Palace
Amalienborg is four nearly identical Rococo palaces arranged around an octagonal courtyard in the Frederiksstaden district — and the Danish Royal Family has lived here since 1794, when Christiansborg burned down and the royals needed somewhere quickly. Architectural historians consider it the finest Rococo complex in Denmark, possibly in all of northern Europe, and it is on Denmark's tentative list for UNESCO World Heritage status.
Two of the four palaces house the royal apartments of King Frederik X and Queen Mary; a third is the official guest wing; and the fourth has been a museum since 1994. The museum preserves royal apartments from the 19th and early 20th centuries — less grand than you might expect, and genuinely interesting for what they reveal about how the monarchy actually lived day to day. The equestrian statue of Frederik V at the centre of the courtyard was made by French sculptor Jacques Saly and took twenty years to complete.
The Changing of the Guard happens daily at noon and is worth timing your visit around. What many visitors miss is that the guards march from Rosenborg Castle through the city streets to reach Amalienborg — the march through town is often more interesting than the ceremony itself. As places to visit in Copenhagen go, the courtyard is free and the museum is reasonably priced; both are worth it.