  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.
