During his time as full professor of mathematics at the University of Vienna from 1903 to 1935, Wilhelm Wirtinger had a great influence on Austrian mathematics in the first third of the twentieth century. He himself had already read writings by Bernhard Riemann during his school days. However, it was Klein who gave Wirtinger the impetus to research Abelian functions during his stay in Göttingen in the summer semester of 1889. After his return to Vienna, Wirtinger specialized in theta functions, which were used to invert Abelian integrals. In response to the task set by Klein for the 1895 Beneke Foundation Prize, Wirtinger wrote a book about these functions as such. He received the prize and was appointed associate professor at the University of Innsbruck in the same year. As early as 1896 he was promoted to full professor there, from where he was called to Vienna in 1903. He also invested a lot of time and energy in publication projects initiated by Klein. Wirtinger was co-editor of the analysis volume for the Encyklopädie and wrote several articles for it. Together with Max Noether he also edited the supplements to Riemann’s collected works.