Levy's N-parameter Brownian motion in d-dimensional space is denoted by W(N,d). Using uniform partitions and a Vitali-type variation, Berman recently extended to W(N,1) a classical result of Le'vy concerning the relation between Wl1 1) and 2-variation. With this variation W(N,d) has variation dimension 2N with probability one. An appropriate definition of weak variation is given using powers of the diameters of the images of sets which satisfy a parameter of regularity. A previous result concerning the Hausdorff dimensions of the graph and image is used to show the weak variation dimension of W(N,d) is 2N with probability one, extending the result for W(ll) of Goffman and Loughlin. If unrestricted partitions of the domain are used, the weak variation dimension of a function turns out to be the same as the Hausdorff dimension of the image. In a recent paper by Gioffman and Loughlin [31 strong and weak variations were defined, and the strong and weak variation dimensions of Browrnian motion were shown to be 2 with probability one. The definitions and proofs given in that paper do not immediately generalize to multiparameter Brownian motion. The purpose of this paper is to find appropriate definitions of strong and weak variations and, using a previous result on the Hausdorff dimensions of Brownian motion (see the appendix for a summary of the proof), to show for N-parameter Brownian motion in d-dimensional space, both the strong and weak variation dimensions are 2N with probability one. The distinction between the strong and weak variations becomes more pronounced in higher dimensions. In N parameters the strong variation is defined in terms of the vertices of rectangles and the weak variation in terms of the oscillation on the rectangle, whereas in the one variable case both the strong and weak variations may be defined in terms of the oscillation on an interval. Presented to the Society May 3, 1973; received by the editors July 27, 1973. AMS (MOS) subject classifications (1970). Primary 6OG17. Copyight i) 1974, American Mathematical Society