We show how the Szekeres form of the line element is naturally adapted to study Penrose limits in classical string backgrounds. Relating the old colliding wave problem to the Penrose limiting procedure as employed in string theory we discuss how two orthogonal Penrose limits uniquely determine the underlying target space when certain symmetry is imposed. We construct a conformally deformed background with two distinct, yet exactly solvable in terms of the string theory on R–R backgrounds, Penrose limits. Exploiting further the similarities between the two problems we find that the Penrose limit of the gauged WZW Nappi–Witten universe is itself a gauged WZW plane-wave solution of Sfetsos and Tseytlin. Finally, we discuss some issues related to singularity, show the existence of a large class of non-Hausdorff solutions with Killing–Cauchy horizons and indicate a possible resolution of the problem of the definition of quantum vacuum in string theory on these time-dependent backgrounds.