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A-Hús
Building houses from scraps didn't mean that artistic impulses were squashed - check out A-Hús, from 1906, one of the city's finest examples of wood-and-tin architecture. Its tall turrets are topped by swirling arabesques, and the wooden struts supporting the balconies are carved with whales.
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Alþingi
Iceland's first parliament, the Alþingi, was created at Þingvellir in AD 930. After losing its independence in the 13th century, the country gradually won back its autonomy, and the modern Alþingi moved into the current basalt building in 1881; a stylish glass-and-stone annexe was completed in 2002. You're welcome to attend sessions (h Mon, Tue & Wed, Thu Oct-May) when parliament is sitting.
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Árbæjarlaug Geothermal Pool
It's a step out of town, but the slickly designed Árbæjarlaug Geothermal Pool is well known as the best family pool: it's half inside and half outside, and there are lots of watery amusements (slides, waterfalls and massage jets) to keep the kids entertained. Take bus 19.
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Árbæjarsafn
Quaint old buildings have been uprooted from their original sites and rebuilt at the open-air Árbæjarsafn, a kind of zoo for houses, 4km from the city centre. Alongside the 19th-century homes are a turf-roofed church, and various stables, smithies, barns and boathouses - all very picturesque. There are summer arts-and-crafts demonstrations, and it's a great place for kids to let off steam. Take bus 12.
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Árni Magnússon Institute
Árni Magnússon Institute Árni Magnússon devoted his life to saving Icelandic manuscripts. If you're interested in seeing them for yourself, contact the instute.
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Ásgrímur Jónsson Museum
If you want to see more of Ásgrímur Jónsson's work than is found at the National Gallery, the National Gallery also owns the Ásgrímur Jónsson Museum, open by appointment only.
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Ásmundur Sveinsson Sculpture Garden
There's something immensely tactile about Ásmundur Sveinsson's monumental concrete creations - see for yourself in the Ásmundur Sveinsson sculpture garden outside the rounded, white Ásmundarsafn.
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Austurvöllur
At the heart of the city, grassy Austurvöllur was once part of first settler Ingólfur Arnarson's hay fields. Today it's a favourite spot for lunchtime picnics and summer sunbathing, and is sometimes used for open-air concerts and political demonstrations. The statue in the centre is of Jón Sigurðsson, who led the campaign for Icelandic independence.
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Culture House
Creeping into the darkened rooms of the Culture House is a true thrill for saga lovers. A permanent exhibition covers saga history: from a Who's Who of Norse gods to a fascinating account of Árni Magnússon, who devoted his life to saving Icelandic manuscripts, and died of a broken heart when his Copenhagen library went up in flames. Two hushed display rooms contain the original vellums.
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Dómkirkja
Compared to the sky-scraping hulk of Hallgrímskirkja, Iceland's main cathedral, Dómkirkja is a modest affair, but it played a vital role in the country's conversion to Lutheranism. The current building (from 1848) is small but perfectly proportioned, its plain wooden interior animated by glints of gold.
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Einar Jónsson Museum
If the Einar Jónsson sculpture garden appeals to your inner Goth, you'll find gleaming white-marble sculptures on similar themes inside the fascinating Einar Jónsson Museum. The building itself was designed by the artist and contains his austere penthouse flat, with unusual views over the city.
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Einar Jónsson Sculpture Garden
Einar Jónsson (1874-1954) is Iceland's foremost sculptor, famous for his intense symbolist works. Chiselled allegories of Hope, Earth, Spring and Death burst from basalt cliffs, weep over naked women, sprout wings and slay dragons. For a taster, the Einar Jónsson sculpture garden behind the Einar Jónsson Museum contains 26 bronze casts; they're particularly effective at dusk.
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Fálkahús
Some of the city's oldest houses line Ingólfstorg square. Fálkahús has a particularly interesting history - it's where Icelandic falcons were kept before being shipped off to Europe's noblemen.
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Family Fun Park & Zoo
The Family Fun Park & Zoo is the city's only attraction especially for (youngish) children. Don't expect lions and tigers; think seals, foxes and farm animals with slightly dismal enclosures. The family park section is jolly, with a mini-racetrack, child-size bulldozers, a giant trampoline, boats and kids' fairground rides. Future plans include an Imax cinema and an aquarium, but at the time of writing building work hadn't started.
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Fríkirkjan í Reykjavík
The old town's mid-18th-century houses demonstrate the Icelandic talent for adaptation. In a country devoid of building materials, most are made from driftwood (which floated from Siberia and South America) and covered in sheets of corrugated tin to protect them from the elements. Even churches, such as the Fríkirkjan í Reykjavík, were made the same way. By happy chance, this light construction method also makes the buildings pretty earthquake-proof.
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Hallgrímskirkja
Reykjavík's most attention-seeking building is the immense concrete church Hallgrímskirkja, star of a thousand postcards and visible from 20km away. For an unmissable view of the city, make sure you take an elevator trip up the 75m-high tower.
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Ingólfstorg
Popular with skateboarders, the stone square Ingólfstorg is notable for its billowing steam vent, where pent-up geothermal energy finds a release.
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Laugar Geothermal Spa
The five-star Laugar Geothermal Spa is attached to Laugardalslaug Geothermal Pool and offers delicious ways to pamper yourself. There are six themed saunas and steam rooms, a vast and well-equipped gym, and beauty and massage clinics with soothing treatments (detox wraps, facials and hot-stone therapies).
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Laugardalslaug Geothermal Pool
Laugardalslaug Geothermal Pool is the largest pool in Iceland, with the best facilities: an Olympic-size indoor pool, an outdoor pool, four hot pots and a whirlpool, a steam bath, and a curling 86m water slide. Take bus 14.
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Laugardalur
Laugardalur was once the main source of Reykjavík's hot water supply - the name translates as 'Hot-Springs Valley'. Buses 14, 15, 17, 19 and S2 pass within a few hundred metres of Laugardalur.
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Leifur Eiríksson statue
Gazing proudly into the distance outside Hallgrímskirkja church is a statue of the Viking Leifur Eiríksson, the first European to stumble across America. It was a present from the USA on the 1000th anniversary of the Alþing.
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National Gallery of Iceland
Surreal mud-purple landscapes are intermingled with visions of trolls, giants and dead men walking at the National Gallery of Iceland. Iceland's main art gallery, overlooking Tjörnin, certainly gives an interesting glimpse into the nation's psyche. As well as a huge collection of 19th- and 20th-century paintings by Iceland's favourite sons and daughters (including Ásgrímur Jónsson, Jóhannes Kjarval and Nína Sæmundsson), there are works by Picasso and Munch.
