In this paper, we study almost nef regular foliations. We give a structure
theorem of a smooth projective variety $X$ with an almost nef regular foliation
$\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with
rationally connected fibers such that $\mathcal{F}$ is a pullback of a
numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a
relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary,
an almost nef tangent bundle of a rationally connected variety is generically
ample. For the proof, we generalize Fujita's decomposition theorem. As a
by-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sum
of a hermitian flat vector bundle and a generically ample reflexive sheaf for
any algebraic fiber space $f : X \rightarrow Y$. We also study foliations with
nef anti-canonical bundles.