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We give a simple criterion for slope stability of Fano manifolds $X$ alongdivisors or smooth subvarieties. As an application, we show that $X$ is slopestable along an ample effective divisor $D\subset X$ unless $X$ is isomorphicto a projective space and $D$ is a hyperplane section. We also givecounterexamples to Aubin's conjecture on the relation between the anticanonicalvolume and the existence of a K\"ahler-Einstein metric. Finally, we considerthe case that $\dim X=3$; we give a complete answer for slope (semi)stabilityalong divisors of Fano threefolds.

Towards a criterion for slope stability of Fano manifolds along divisors