1. National Museum of Iceland: Visit this museum to explore the country's history from settlement to modern times. Exhibitions include artifacts, archaeological findings, and displays showcasing Iceland's cultural heritage.

2. Reykjavik City Museum: Discover the city's history and evolution through interactive exhibitions, multimedia displays, and historical objects. The museum consists of several sites, including Árbær Open Air Museum, Reykjavik Maritime Museum, and Reykjavik Museum of Photography.

3. Settlement Exhibition: Located in downtown Reykjavik, this museum offers insight into Iceland's Viking past. Explore the remains of a Viking longhouse and learn about the early settlement period through interactive exhibits.

4. The National Gallery of Iceland: While primarily an art museum, the National Gallery also features exhibitions focusing on Iceland's art history. Dive into the rich cultural heritage of the country through various visual art forms.

5. Museum of Icelandic Sorcery and Witchcraft: Delve into the fascinating world of magic, folklore, and witchcraft in Iceland. This quirky museum provides insight into the country's supernatural beliefs and practices throughout history.

6. The Culture House: Immerse yourself in Icelandic heritage and art at this museum. Discover a wide range of exhibits, including historical artifacts, medieval manuscripts like the famous "Book of Icelanders," and contemporary Icelandic art.

7. The Icelandic Punk Museum: Trace the history of punk rock in Iceland at this unique museum located in an underground bunker. Explore the exhibits showcasing the beginnings and impact of the punk movement in the country.

8. The Saga Museum: Experience Iceland's sagas, which depict the country's medieval history, through lifelike wax figures. Walk through different scenes and learn about the events and characters that shaped Icelandic saga literature.

Note: It's always recommended to check the opening hours and any changes in operation due to COVID-19 before visiting the museums.