  In this paper, we study smooth complex projective varieties $X$ such that
some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We
prove that such varieties are rationally connected. We also classify the
following two cases. If $T_X$ is strictly nef, then $X$ isomorphic to the
projective space $\mathrm{P}^n$. If $\bigwedge^2 T_X$ is strictly nef and if
$X$ has dimension at least $3$, then $X$ is either isomorphic to $\mathrm{P}^n$
or a quadric $\mathrm{Q}^n$.
