  In this paper we study asymptotic directions in the tangent bundle of the
moduli space ${\mathcal M}_g$ of curves of genus $g$, namely those tangent
directions that are annihilated by the second fundamental form of the Torelli
map. We give examples of asymptotic directions for any $g \geq 4$. We prove
that if the rank $d$ of a tangent direction $\zeta \in H^1(T_C)$ (with respect
to the infinitesimal deformation map) is less than the Clifford index of the
curve $C$, then $\zeta$ is not asymptotic. If the rank of $\zeta$ is equal to
the Clifford index of the curve, we give sufficient conditions ensuring that
the infinitesimal deformation $\zeta$ is not asymptotic. Then we determine all
asymptotic directions of rank 1 and we give an almost complete description of
asymptotic directions of rank 2.
