By the old harbour in Reykjavík, Hafnarhúsið hosts a rotating exhibitions of the work of Icelandic artist Erró and temporary exhibitions often showcase other local artists.
Reykjavik Art Museum is the largest visual art institution in Iceland. It occupies three locations in Reykjavík; Hafnarhús by the old harbour (64.1490°N 21.9406°W), Kjarvalsstaðir by Klambratún (64.1378°N 21.9135°W) and Ásmundarsafn in Laugardalur (64.1416°N 21.8853°W).
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Ljósmyndasafn Reykjavíkur
A very small museum with a nice library and reading room where you can find some older (but good) books about photography and current and past issues of photography magazines. It also has a huge collection of Icelandic photographs.
The Settlement Exhibition
Run by the Reykjavík City Museum, this exhibition in central Reykjavík was built around the oldest archaeological ruins in Iceland. As the name indicates, these ruins date to around the year 870. This interactive exhibitions brings you the early history of the area that today forms central Reykjavík.
A small park (or square, depending on definitions) in the heart of Reykjavík. It's many locals' favorite place to spend sunny days, either at one of the cafés lining the north of the square or simply having a picnic on the grass. The parliament and the national cathedral both stand by Austurvöllur.
On the southern edge of Austurvöllur is a small building of hewn stone, but don't let its size fool you. This is the building of the Icelandic parliament, known as Alþingi. The institution has in fact long since outgrown the building which was built in 1881 for a nation of a little over 60,000. Today the upper floors of most houses on the north and west sides of the park also house parliamentary offices. The Alþingi building today houses only the debating chamber of the unicameral institution and the party meeting rooms. When Alþingi is in session it is possible to go up to the viewing platforms and watch the debates, otherwise only groups affiliated with educational institutions are permitted to tour the inside of the building.
Pönksafn Íslands
A small museum devoted to the history of punk music in Iceland, located in a former underground public toilet facility. In the final room you can listen to Icelandic punk music through headphones, and there are guitars and leather jackets to try on and pose with.
With the exception of Húsavík in the north, Reykjavík is actually one of the very best places to go whale watching in Iceland. Whales frequently come into Faxaflói, the large bay which Reykjavík sits by and on a typical trip of around 3 hours you can almost be guaranteed to see at least some minke whales and possibly even a humpback. The companies offering whale watching mostly occupy a small area in the old harbour called Ægisgarður, close to the whaling ships. All sail out to the same bay but since conditions there change make sure you are on a good ship.