In this thesis we consider questions arising in Gromov-Witten theory, quantum cohomology and mirror symmetry. The first two chapters deal with Gromov-Witten theory and derived categories for moduli spaces of stable curves of genus zero with n marked points. In the third chapter we consider Landau-Ginzburg models for odd-dimensional quadrics. In the first chapter we study moduli spaces of stable maps with target being the moduli space of stable curves of genus zero with n marked points, and curve class being a class of a boundary curve. An explicit formula for the respective Gromov-Witten invariants is given. In the second chapter we consider inductive constructions of semi-orthogonal decompositions and exceptional collections in the derived category of moduli spaces moduli spaces of stable curves of genus zero with n marked points based on a nice presentation of these spaces as consecutive blow-ups due to Keel. In the third chapter we give an ad hoc partial compactification of the standard Landau-Ginzburg potential for an odd-dimensional quadric, and study its Gauss-Manin system in the case of three dimensional quadrics. We show that under some hypothesis this Landau-Ginzburg potential would give a Frobenius manifold isomorphic to the quantum cohomology of a three dimensional quadric
