A ten-minute ferry from the old harbour takes you to Viðey, a low green island in the bay that was inhabited from Viking times until the 1940s and is now a public park. Viðey Church (1774) is one of the oldest stone buildings in Iceland; next to it stands Viðey House, the country's first proper masonry residence.
Walking paths circle the island in about two hours and pass the ruined village, the bird cliffs, and the site of the medieval monastery. The Yoko Ono Imagine Peace Tower, an outdoor light sculpture dedicated to John Lennon, stands at the south end. Check the ferry timetable before you set out: services are seasonal and weather-dependent.