It is safe to say that Jóhannes Kjarval (1885-1972) is the single biggest name in Icelandic painting. Kjarvalsstaðir hosts a collection of his work, as well as hosting other temporary exhibitions.
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Klambratún is a park just east of the city centre on an area which remained farmland while the city was built up around it. The area was later converted into one of the largest public parks in the city and often hosts various events. One of the houses of the Reykjavík Art Museum, Kjarvalsstaðir, is inside the park.
The city's oldest and only indoor pool (with outdoor hot-pots), located in the city centre. Has a more municipal feel than the other pools, but has a very central location.
This can't miss attraction towers over the city on top of a hill. In front is a statue of Leif Ericsson (Leifur Eiríksson in Icelandic), the Norse explorer who sailed to North America in the 10th century. The United States gave this statue to Iceland in 1930, in honor of the 1,000th anniversary of the Althingi, the Iceland parliament.
The Pearl
An iconic building on top of a wooded hill called Öskjuhlíð, to the southeast of the city centre. Perlan is built on top of five hot water storage tanks and offers fantastic views of the entire city from a viewing platform open to the public and from a rotating restaurant at the top. If the restaurant is too expensive for you (it is for most), there is also a small cafeteria on the same floor as the viewing platform. For the price of admission, one can view and interact with exhibits related to the geology, geological history, and natural history of Iceland. There are some shows related to these topics as well covered with the tickets.
Dómkirkjan í Reykjavík
The church beside the parliament is Reykjavík Cathedral, the mother church of the Church of Iceland, the country's national church and part of the Lutheran denomination. Similarly deceptive in size, it has been beautifully renovated both inside and out to reflect its original 18th-century architecture.
Borgarleikhúsið
Like the national theatre, the city theatre puts on a mix of new Icelandic plays and highlights of international theatre.