Let X be a complex-projective contact manifold with $b_2(X)=1$. It has long been conjectured that X should then be rational-homogeneous, or equivalently, that there exists an embedding $X \to {\mathbb P}^n$ whose image contains lines.

We show that X is covered by a compact family of rational curves, called ‘contact lines’, that behave very much like the lines on the rational-homogeneous examples: if $x \in X$ is a general point, then all contact lines through x are smooth, no two of them share a common tangent direction at x, and the union of all contact lines through x forms a cone over an irreducible, smooth base. As a corollary, we obtain that the tangent bundle of X is stable.