In 1977 Sigrún graduated from the department of graphic design at The Icelandic College of Arts and Crafts. Sigrún has worked as an artist since 1978. She has exhibited her work extensively, both in solo-exhibitions and participating in exhibitions in Iceland and abroad, such as Scandinavia, Poland, Germany, USA, S-Korea, Taiwan and Japan. Alongside her art, Sigrún has worked as a writer and has published numerous books for children. Her first book, Allt í plati (Just Joking) was published in 1980. She illustrates all her books and has also illustrated a number of books for other authors, such as Guðrún Helgadóttir, Magnea frá Kleifum and her brother, Þórarinn Eldjárn. He has also written poetry for her picture books. Sigrún and Þórarinn have received various awards for their books, and they were nominated for the Nordic Children's Book Prize in 1998. Sigrún has also received an honorary listing from the H.C. Andersen Prize. Apart from writing books, Sigrún has also written television manuscripts for The Icelandic National Broadcasting Service.