The most recognisable building in Iceland, named for the seventeenth-century poet Hallgrímur Pétursson and designed to echo the basalt columns you see all over the country. The architect Guðjón Samúelsson started work in the 1940s; it took forty-one years to finish. The interior is famously stark — white concrete, plain wood, an enormous pipe organ — and pleasantly quiet compared to the queue outside.
Pay the small fee to take the lift up the 75-metre tower for the best in-town view: coloured rooftops fanning out toward the harbour and Mount Esja behind. The statue of Leif Eriksson on the lawn was a 1930 gift from the United States.