  In this paper, we develop the theory of singular hermitian metrics on vector
bundles. As an application, we give a structure theorem of a projective
manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth
fibration $X \to Y$ to a flat projective manifold $Y$ such that its general
fiber is rationally connected. Moreover, by applying this structure theorem, we
classify all the minimal surfaces with pseudo-effective tangent bundle and
study general non-minimal surfaces, which provide examples of (possibly
singular) positively curved tangent bundles.
