In this thesis we study the space of lines and conics on special varieties, and we apply this study to address relevant problems concerning the geometry of Fano varieties. In the first part of this thesis we study general codimension 2 linear sections of the Grassmannians G(2, 5) and G(2, 6) (in the Plücker embedding). We give a description of their spaces of lines passing through any given point. As an application we show that these Fano manifolds are not weakly 2-Fano, completing the classification of weakly 2-Fano manifolds of high index, initiated by Araujo and Castravet. In the second part, we study conic-connected manifolds. We prove that the space W_x,y of conics on a conic-connected manifold X passing through two general points x,y in X is smooth, and we define a natural polarization on this space. Relating this study with the study of minimal pointed rational curves by de Jong and Starr, we give a formula for the canonical bundle of W_x,y in terms of the second Chern character of X and the first Chern class of our polarization. We conclude that W_x,y is Fano if X is weakly 2-Fano.