  Ross and Thomas introduced the concept of slope stability to study
K-stability, which has conjectural relation with the existence of constant
scalar curvature K\"ahler metric. This paper presents a study of slope
stability of Fano manifolds of dimension $n\geq 3$ with respect to smooth
curves. The question turns out to be easy for curves of genus $\geq 1$ and the
interest lies in the case of smooth rational curves. Our main result classifies
completely the cases when a polarized Fano manifold $(X, -K_X)$ is not slope
stable with respect to a smooth curve. Our result also states that a Fano
threefold $X$ with Picard number 1 is slope stable with respect to every smooth
curve unless $X$ is the projective space.
