A submanifold of an affine space is "totally skew" if it has no pairs of parallel or intersecting tangent lines at distinct points. Given a manifold M^n , what is the least dimension of an affine space that can contain M as a totally skew manifold? Even for a disk, this is a hard question, closely related to the generalized vector field problem, non-singular bilinear maps and the immersion problem for real projective spaces.