Abstract: Let V be a finite dimensional vector space and let G be a subgroup of GL(V). In general the quotient space V/G of G-orbits can be as bad as you want it to be. There are some fairly broad classes of solvable groups G however for which V/G admits a useful, though incomplete,description. In this talk I will use a number of examples to illustrate a method that obtains the following.(i) A finite partition of V into invariant algebraic subsets W.(ii) For each W, an explicit algebraic subset S of W that is a cross-section for the orbits in W.(iii) A fiber bundle structure P: W --> S for which P-1(v) = Gv, v in S .Specific theorems assert that this method can be carried out for certain classes of solvable groups. If it seems desirable, then some indication of the applicability of these results to harmonic analysis may be given. I will finish by discussing the difficulties in carrying this method out for other classes of solvable groups.