The Square element graph over a semigroup S is a simple undirected graph Sq(S) whose vertex set consists precisely of all the non-zero elements of S, and two vertices a,b are adjacent if and only if either ab or ba belongs to the set {t 2 :t∈S}∖{1}, where 1 is the identity of the semigroup (if it exists). In this paper, we study the various properties of Sq(S). In particular, we concentrate on square element graphs over three important classes of semigroups. First, we consider the semigroup Ω n formed by the ideals of Z n . Afterwards, we consider the symmetric groups S n and the dihedral groups D n . For each type of semigroups mentioned, we look into the structural and other graph-theoretic properties of the corresponding square element graphs. © 2019 The Korean Institute of Communications and Information Sciences (KICS)