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This paper develops a vector theory of cross-phase modulation (XPM) in optical fibers and use it to investigate the impact of polarization-mode dispersion (PMD) on the crosstalk induced by XPM in wavelength-division multiplexed lightwave systems. Under certain reasonable approximations, the theory permits us to obtain an analytic expression for the amplitude of probe fluctuations induced by a copropagating pump channel through XPM. We use this expression to calculate the average level of XPM-induced crosstalk together with its variance for several dispersion maps. We show that PMD not only reduces the crosstalk on average, but also impacts the efficiency of a commonly used polarization-interleaving technique.