A method of calculating average moments, to an arbitrary order m, of the principal components of the gyration tensor of a Gaussian molecule imbedded in any k-dimensional space is presented. These average moments are then used in generalized shape parameters Am of degree m≤k, which measure the asymmetry of the shapes of Gaussian molecules. Simple formulas for A2, A3, and the average moments up to third order are given. Explicit expressions for A2 and A3 for linear and circular chains, regular stars with infinitely long arms, double rings of a large number of beads, and combs with many side chains are obtained, and the shape characteristics of these molecules are discussed. It is found that Gaussian molecules are, on the average, prolate rather than oblate even in an infinite dimensional space, with the exception of regular stars with densely radiated long arms which exhibit perfect symmetry. The problem of analytic characterizations of shape asymmetry of Gaussian molecules or non-self-avoiding random walks of any kind is thus solved in complete generality.