An analysis of the distribution function of the radius of gyration for Gaussian molecules of arbitrary complexity imbedded in d-dimensional space is carried out. An explicit expression for the coefficients in an exact series expansion of the distribution function is obtained as finite sums of terms involving only the eigenvalues of the connectivity matrix. For odd d, a recursive method is developed for evaluating the distribution function as a sum of simple integrals. For even d, a finite-sum representation exists. The leading term of the asymptotic series for the distribution function for large values of the argument is analytically evaluated for any Gaussian molecule of arbitrary length. Moments of the distribution are given in terms of either the eigenvalue spectrum or the coefficients of the eigenpolynomial of the connectivity matrix. For a special class of molecules, the limiting distribution functions are found to share a generic form which can be represented by a double series. The length dependence of the 3D distribution function for linear chains and the dimensionality effect on the breadth of the limiting distributions for both circular and linear chains are investigated numerically. It isfound that higher dimensionality tends to broaden the distribution.