For an infinite set X,denote by Ω(X) the semigroup of all surjective mappings from X to X.We determine Green’s relations in Ω(X),show that the kel (unique minimum ideal) of Ω(X) exists and determine its elements and cardinality.For a countably infinite set X,we describe the elements of Ω(X) for which the D-class and J-class coincide.We compare the results for Ω(X) with the corresponding results for other transformation semigroups on X.