%0 Journal Article %T Twistor Approach to String Compactifications: a Review %+ Laboratoire Charles Coulomb (L2C) %A Alexandrov, Sergey %Z 98 pages %< avec comité de lecture %Z L2C:11-255 %@ 0370-1573 %J Physics Reports %I Elsevier %V 522 %N 1 %P 1-57 %8 2012-09-17 %D 2012 %Z 1111.2892 %R 10.1016/j.physrep.2012.09.005 %Z Physics [physics]/High Energy Physics - Theory [hep-th]Journal articles %X We review a progress in obtaining the complete non-perturbative effective action of type II string theory compactified on a Calabi-Yau manifold. This problem is equivalent to understanding quantum corrections to the metric on the hypermultiplet moduli space. We show how all these corrections, which include D-brane and NS5-brane instantons, are incorporated in the framework of the twistor approach, which provides a powerful mathematical description of hyperkahler and quaternion-Kahler manifolds. We also present new insights on S-duality, quantum mirror symmetry, connections to integrable models and topological strings. %G English %L hal-00642280 %U https://hal.science/hal-00642280 %~ CNRS %~ L2C %~ MIPS %~ UNIV-MONTPELLIER %~ UM-2015-2021