It is a well known fact that a general hypersurface of degree d in projective n-space is rationally connected if d is at most n, but contains very few curves if d is larger than n. More generally let X be a smooth projective variety and H a hypersurface of X such that K_X+H is anti-ample, then by the adjunction formula and a classical result of Kollar-Miyaoka-Mori we know that H is rationally connected. In a recent project we use the minimal model program as well as other techniques in birational geometry to study further how the behavior of rational curves on X as well as the positivity of -(K_X + H) and H influence the behavior of rational curves in H. In this talk I will present several results and examples of this kind. In particular we will see criteria for uniruledness and rational connectedness of H. Speaker(s): Yuan Wang (University of Utah)