In this talk, we discuss the geometry of a projective klt pair (X, D) with the nef anti-log canonical divisor and its maximally rationally connected (MRC) fibrations. First I explain a relation of the (numerical) Kodaira dimensions of the anti-log canonical divisors on $X$ and on a general fiber of MRC fibrations. This result generalizes Hacon-McKernan's question (which was recently proved by Ejiri-Gongyo), and also can be seen as a generalization of the well-known fact that any weak Fano varieties are rationally connected. Moreover I give the structure theorem for MRC fibrations of X, that is, MRC fibrations can be chosen to be a locally trivial morphism to a smooth projective variety Y of Calabi-Yau type. This is a a generalization of Cao-Horing's structure theorem to klt pairs. Our proof is based on analytic methods, including analytic positivity of direct images, certain flatness, and foliations. This talk is based on a joint work with Frederic Campana and Junyan Cao.