A low concrete-and-glass building by the architects Studio Granda, finished in 1992, the City Hall (Ráðhús Reykjavíkur) sits half on land and half over Tjörnin lake — wet basalt columns rising straight from the water on the south side, a glass cafeteria on the lakefront facing the city.
The cafe inside is open to the public and has a free large-scale relief map of Iceland that is worth a look if you plan to head out into the country. Free WiFi, decent espresso, and a useful refuge in bad weather.