A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP: The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph (Mutzel (Ed.), Graph Drawing 2001, Lecture Notes in Computer Science, Springer, Berlin; Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC-2001)). These results implied that the recognition problem lies in NEXP: In the present paper we improve this by showing that the recognition problem for string graphs is in NP; and therefore NP-complete, since Kratochvı´l showed that the recognition problem is NP-hard (J. Combin Theory, Ser. B 52). The result has consequences for the computational complexity of problems in graph drawing, and topological inference. We also show that the string graph problem is decidable for surfaces of arbitrary genus.