Coherent sheaves on Calabi Yau manifolds Picard Fuchs equations and potential functions Abstract: We investigate the deformation theory of Calabi Yau threefolds simply connected complex projective manifolds with trivial canonical bundle together with geometric objects on the Calabi Yau threefold. The geometric objects in question are curves surfaces special coherent sheaves and rank two vector bundles with a section vanishing in codimension two.<br><br>These deformation problems are related to the study of Picard Fuchs equations. Physicists suggest that a holomorphic potential function the critical locus of which is the space of unobstructed deformations appears as a solution of the Picard Fuchs equation.<br>In this thesis Picard Fuchs equations for Calabi Yau threefolds<br>appearing as complete intersections of codimension two in projective spaces are studied. In addition Picard Fuchs equations for pairs of a Calabi Yau threefold and either a divisor or a curve on the threefold are examined. We construct Picard Fuchs equations using Griffiths Dwork reduction.<br><br>Based on the work of Jockers and Soroush we give rigorous mathematical foundations for deriving Picard Fuchs operators in various cases in particular for the quintic threefold together with a special divisor.<br>Furthermore we initiate a theory of triples consisting of a threefold with two divisors meeting transversally along a curve and set up Picard Fuchs operators for this situation. The thesis furthermore contains some SINGULAR programmes for explicit calculations