Nobelmuseet presents the history of the Nobel Prizes and their recipients, with a focus on the intellectual and cultural aspects of invention. It's a polished, contemporary space with fascinating displays, including short films on the theme of creativity, interviews with laureates like Ernest Hemingway and Martin Luther King, and cafe chairs signed by the visiting prize recipients (flip them over to see!).
The museum is housed in the Börsen building – the old Stock Exchange – which forms the north side of Stortorget, the historic main square in Gamla Stan. Free, highly recommended guided tours of the museum (35 minutes) run several times daily; see the website for the current schedule.