Tawney was born Leonora Agnes Gallagher in Lorain, Ohio, in 1907. She studied drawing, sculpture, and weaving with well-known artists at the University of Illinois in Champaign-Urbana (1943-45), the Institute of Design in Chicago (1946-48), and at the Penland School of Crafts in North Carolina where she became a weaver (1954). Throughout her prolific career, Tawney has had numerous one-person exhibitions and participated in many group exhibitions, including the renowned landmark international Fiberworks at the Cleveland Museum of Art in 1977. Her work is represented in many major museums, including four pieces in the Cleveland Museum of Art, and she has received numerous awards, including the American Craft Council's Gold Medal in 1987. Tawney's "flag" is one of the highlights of the museum's contemporary fiber art collection, which includes about 100 works.