If you budget your time, you can visit two of these sites in one day.

The Karen Blixen Museum at Rungsted is the childhood home of the Danish author of Out of Africa and many other works. She returned to Rungsted as an adult and, under the pen name Isak Dinesen, wrote most of her books here. The house is filled with her personal belongings, including many souvenirs of her days in Africa.

Louisiana Museum of Modern Art, in Humlebaek is a haven for art lovers, featuring the work of the world’s great 20th- and 21st-century artists. Known for its innovative exhibitions, both small-scale and large, the museum is surrounded by a pleasant park and sculpture garden.

Helsingoer, better known as Elsinore, is the home of Kronborg Castle, the inspiration for the setting of Shakespeare’s Hamlet. The castle and its rooms are open to visitors.

Frederiksborg Castle at Hillerød is the traditional coronation site for Danish monarchs. Dating from the 16th century, the castle features stunning chambers and ceremonial rooms as well as a museum.

Due west of Copenhagen, this one-time Danish capital offers enough to fill a full day.

Since the Middle Ages, Roskilde Cathedral has been the burial place of Danish monarchs. The red brick Gothic structure, built in the 12th and 13th centuries, has been expanded over the centuries in unusual ways, making it a delight for historians and architecture aficionados alike.

The Viking Ship Museum came about after a cache of Viking ships were discovered submerged in the fjord beside the city. The painstaking excavation and reconstruction of the ships is shown in an excellent video at the museum. The ships themselves, along with a wealth of information about Viking seafaring history and skills, are displayed in the main hall.

Read more about Roskilde.

Choose one for a leisurely day-trip.

Arken Museum for Modern Art in Ishoj is part gallery, part concert hall, with wonderful views of the flat sandy beach and the Bay of Køge.

Køge is a pretty medieval town with a large number of half-timbered buildings.

The Tycho Brahe Museum on the island of Ven celebrates the life and work of Denmark’s ancient astronomer.

If you really want to get out of town, hop a train and cross the beautiful new Øresund Bridge to Mälmo, Sweden.