Reykjavik

The front rises in stepped tiers copying the way basalt columns split. The tower stands 74.5 metres and a lift carries visitors up to look down on the coloured roofs.

A skin of glass honeycomb throws back a different colour as the weather changes. Looked at from inside, the hexagons break the harbour view into separate pieces.

A steel sculpture on the shore looks like a viking ship stripped to its ribs. The artist stated it is not a ship at all but a vessel carrying dreams.

A glass dome caps six hot water tanks and serves as a viewing deck. Inside, an artificial ice cave 100 metres long can be walked through.

Boats heading out to see whales share the quay with the fishing fleet. Warehouses turned into restaurants stand next to the ice houses.

The main shopping street crosses the centre with murals painted down the side lanes. In summer the cars are barred and people walk down the middle.

Outdoor pools filled with geothermal water sit across the city and locals come daily. Washing without clothes before entering is a rule that is strictly kept.

A tenth century house floor found during building work was left where it lay and a museum was raised over it. Screens run around the walls showing the landscape of that time.

A spit at the western edge carries it, and the crossing works only at low tide. Away from the city lights, people come on winter nights for the aurora.

This indoor market opens at the weekend and sells used books, wool sweaters and dried fish together. In one corner fermented shark is handed out on toothpicks to taste.

One unbroken line of history from settlement to the present day. The piece people stand before longest is the carved wooden door from Valthjofsstadur church, where a knight slays a dragon. It neighbours the university campus, walkable from downtown.

City Hall is built half out over the water of Tjornin pond. Anyone can walk into the ground floor, where a large relief map models the whole country. Ducks and swans crowd the pond, and there are always children arriving with bread.

A white timber house from 1909 where Reagan and Gorbachev held their 1986 summit. The interior is not open, so you see it from the seaside lawn. A British diplomat once asked for it to be sold, blaming a ghost.

A small island you cross to on a ferry from the old harbour. Yoko Ono's Imagine Peace Tower shines a column of light into the sky from 9 October, John Lennon's birthday, to 8 December. There are no cars, so you walk the grassy paths.

A geothermal lagoon opened in 2021 on the Kopavogur shore. Its edge is built to look as if it runs into the sea, and on clear days Snaefellsjokull shows in the distance. You enter between turf walls and follow a set circuit through a cold plunge and a sauna.

A city beach made by trucking golden sand into a cove beside the domestic airport. Hot water is fed toward the sea so the lagoon inside the wall warms a little. Even in winter some locals swim out and then run to the hot tub.

Twenty-six bronzes fill the garden behind the sculptor's home right next to Hallgrimskirkja. The garden is free and you enter through the wall gate. It is a calm spot to rest away from the crowds around the church.

An open-air museum around the old Arbaer farm, filled with houses moved out of central Reykjavik to make room for building work. Turf-roofed homes and wooden shops line up like a street. In summer staff in period dress show the old household chores.

The mountain wall across the bay, in sight from almost every street. A bus reaches the trailhead and the summit round trip takes four to five hours. Many turn back at the big boulder Steinn, since the last stretch above it is a scramble using your hands.

The street sloping from Hallgrimskirkja down into town is painted in rainbow stripes. They were laid for the 2015 Pride festival and stayed. Workshops and wool shops line both sides, so it is the natural way down from the church.

A green belt following the Ellidaar river through the eastern part of the city. Salmon run up this river inside the city limits, so in summer you pass people holding fishing rods. Walking and cycling paths wind past small falls and birch woods.

A small lawn square fronted by the stone Parliament House of 1881 and the cathedral beside it. In 2009, during the financial crisis, protesters banged pots and pans here. On sunny days locals lie on the grass with their lunch.

A 198-metre fall at the head of Hvalfjordur, ranked second in height among the nation's waterfalls. Drive just over one hour from the capital, then hike three to four hours there and back to look down on it. The trail passes a cave and crosses the river.

An open lift lowers you into the empty magma chamber of a volcano that has not erupted in 4000 years. It is 213 metres deep, walls streaked red and yellow by minerals. The site is a 30-minute drive from town, then a long hike over lava from parking.