A graph G with vertex set \(\{v_1,v_2,\ldots ,v_n\}\) is an intersection graph of segments if there are segments \(s_1,\ldots ,s_n\) in the plane such that \(s_i\) and \(s_j\) have a common point if and only if \(\{v_i,v_j\}\) is an edge of G. In this expository paper, we consider the algorithmic problem of testing whether a given abstract graph is an intersection graph of segments. It turned out that this problem is complete for an interesting recently introduced class of computational problems, denoted by \(\exists \mathbb {R} \) . This class consists of problems that can be reduced, in polynomial time, to solvability of a system of polynomial inequalities in several variables over the reals. We discuss some subtleties in the definition of \(\exists \mathbb {R} \) , and we provide a complete and streamlined account of a proof of the \(\exists \mathbb {R} \) -completeness of the recognition problem for segment intersection graphs. Along the way, we establish \(\exists \mathbb {R} \) -completeness of several other problems. We also present a decision algorithm, due to Muchnik, for the first-order theory of the reals.