A solvmanifold (nilmanifold) is the homogeneous space of a connected solvable (nilpotent) Lie group. A theorem of A. I. Malcev [3] states that a nilmanifold can always be expressed as the quotient of a nilpotent Lie group by a discrete subgroup. From this it follows easily that nilmanifolds are parallelizable. Since the Klein bottle is a solvmanifold (see, for example, [1]), we cannot hope to prove solvmanifolds parallelizable'. However, we prove in ? 2 that any n-dimensional compact solvmanifold has n independent line element fields; i.e. Mis line element parallelizable. Using this result and the results of [4], we see that the Stief el-Whitney classes of M can be expressed very simply in terms of one dimensional cohomology classes of M. We also prove in ? 2 that the complexification of the tangent bundle of a compact solvmanifold is trivial so that all Pontrjagin classes are zero. In ? 3, we give an example of a compact orientable five dimensional solvmanifold M which is not parallelizable2. This shows that our results are essentially best possible. In fact, Mis also a riemannian flat manifold (a riemannian manifold with curvature and torsion zero) and we prove that the second Stiefel-Whitney class of M is non-zero. This shows that the Stiefel-Whitney classes of a riemannian manifold cannot be expressed in terms of the riemannian curvature. In conclusion, the authors would like to express their gratitude to W. S. Massey for several stimulating conversations.