Spatial surfaces with parallel mean curvature vector play some important roles in general relativity, theory of harmonic maps, as well as in differential geometry. Recently, spatial surfaces in four-dimensional Minkowski space time with parallel mean curvature vector were classified by Chen and Van der Veken [“Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms,” Tohoku Math. J. 61, 1 (2009)]. In this article, we completely classify spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension. The main result states that there exist 16 families of such surfaces. As by-product, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in Minkowski space times of arbitrary dimension.
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