  In this paper, we extend the structure theorem for smooth projective
varieties with nef tangent bundle to projective klt varieties whose tangent
sheaf is either positively curved or almost nef. Specifically, we show that
such a variety $X$, up to a finite quasi-\'etale cover, admits a rationally
connected fibration $X \to A$ onto an abelian variety $A$. For the proof, we
develop the theory of positivity of coherent sheaves on projective varieties.
As applications, we establish some relations between the geometric properties
and positivity of tangent sheaves.
