She received her PhD in 1949 from the University of Texas at Austin under the direction of R.L. Moore. Rudin then taught at Duke University until 1953. It was at Duke that she met her future husband, Walter Rudin (who died in 2010). They relocated to the University of Rochester where she held a visiting position until 1958. The Rudins then moved to the University of Wisconsin-Madison, where Mary Ellen was appointed as a lecturer. In 1971 she was appointed to full professor and became the first Grace Chisholm Young Professor at the university. Later she was named Hilldale Professor.
Rudin retired from the university in 1991 but remained active in research and math department activities for many years. She published approximately 70 research articles and supervised 18 PhD students. In an interview with Don Albers and Constance Reid (College Mathematics Journal, March 1988) Rudin, mother of four, said, "I have never minded doing mathematics lying on the sofa in the middle of the living room with the children climbing all over me. I feel more comfortable and confident when I'm in the middle of things, and to do mathematics you have to feel comfortable and confident." She was a member of the AMS since 1948. See pictures and remembrances of her on the University of Wisconsin-Madison Department of Mathematics site and read more about her at the MacTutor History of Mathematics archive.
(From the American Mathematical Society website, ams.org, 20 March 2013.)