In this paper, we show that if the
tangent bundle of a smooth projective variety is strictly nef, then
it is isomorphic to a projective space; if a projective variety
$X^n$ $(n>4)$ has strictly nef $\Lambda^2 TX$, then it is
isomorphic to $\P^n$ or quadric $\Q^n$. We also prove that on
elliptic curves, strictly nef vector bundles are ample, whereas
there exist Hermitian flat and strictly nef vector bundles on any
smooth curve with genus $g\geq 2$.