In this thesis we prove the infinitesimal Torelli theorem for certain classes of irregularvarieties. Given a compact Kähler manifold, the infinitesimal Torelli problemasks whether the differential of the period map of a Kuranishi family is injective.Unlike the classical Torelli theorem for curves, there is a negative answer for examplefor hyperelliptic curves of genus greater than 2. Nevertheless, the infinitesimalTorelli theorem holds for many other classes of manifolds. Following Green’s prooffor sufficiently ample hypersurfaces in arbitrary varieties, we prove it for smoothample hypersurfaces and more generally complete intersections in general abelianvarieties by reducing it to showing the surjectivity of certain multiplication mapsof vector bundles on the ambient abelian variety. Then we derive numerical conditionsfor such multiplication maps to be surjective giving an effective bound onGreen’s result in this particular case. We also investigate the more general case ofirregular varieties with globally generated cotangent bundle which do not embedinto their Albanese varieties.