The first axiomatisation of set theory was given by Zermelo in his 1908 paper “Untersuchungen über die Grundlagen der Mengenlehre, I” (Zermelo 1908b), which became the basis for the modern theory of sets. This entry focuses on the 1908 axiomatisation; a further entry will consider later axiomatisations of set theory in the period 1920–1940, including Zermelo's second axiomatisation of 1930.