Maria Goeppert Mayer is frequently mentioned as an example of a woman who managed to make significant contributions to science in spite of tremendous obstacles. Robert Sachs, Mayer's first graduate student, has given a personal account of her life and career (PHYSICS TODAY, February 1982, page 46). Joan Dash has written a longer biography of Mayer, focusing on her family life. In this article I look at the distinctive character of Mayer's scientific accomplishments. As a mathematical physicist, she worked on a number of apparently unrelated topics during the years 1930–46. I will focus on her work during this period and show how it prepared her for the nuclear physics research for which she received the Nobel Prize in 1963.