An embedding $f : M \to \mathbb{R}^n$ is called totally skew if for every pair of distinct points $x,y \in M$, the tangent spaces at $f(x)$ and $f(y)$ neither intersect nor contain parallel directions.  Following Ghomi and Tabachnikov, we ask: Given a manifold $M$, what is the smallest dimension $n(M)$ such that $M$ admits a totally skew embedding into $\mathbb{R}^n$?  The answer is known in only three cases: $n(\mathbb{R})=3$, $n(\mathbb{R}^2) = 6$, and $n(S^1)=4$.  In their investigation, Ghomi and Tabachnikov established a deep relationship between totally skew embeddings and the generalized vector field problem, as well as to immersions and embeddings of real projective spaces.  We give a brief overview of their work and discuss our recent progress on the existence problem for totally skew embeddings, which is based on an h-principle technique due to Gromov and Eliashberg.