  The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a
natural generalization of the concept of classical graph distance. For a
connected graph $G$ of order at least 2 and $S\subseteq V(G)$, the Steiner
distance $d_G(S)$ of the set $S$ of vertices in $G$ is the minimum size of a
connected subgraph whose vertex set contains or connects $S$. In this paper, we
introduce the concept of the Steiner (revised) Szeged index ($rSz_k(G)$)
$Sz_k(G)$ of a graph $G$, which is a natural generalization of the well-known
(revised) Szeged index of chemical use. We determine the $Sz_k(G)$ for trees in
general. Then we give a formula for computing the Steiner Szeged index of a
graph in terms of orbits of automorphism group action on the edge set of the
graph. Finally, we give sharp upper and lower bounds of ($rSz_k(G)$) $Sz_k(G)$
of a connected graph $G$, and establish some of its properties. Formulas of
($rSz_k(G)$) $Sz_k(G)$ for small and large $k$ are also given in this paper.
