The birational classification of projective manifolds was developed by F. Severi, R. Hall and D. Lieberman. Employing the Albanese morphism the classification can be made biholomorphic in the non-uniruled case, extending the results of E. Calabi on manifolds with trivial $c_1(X)$. The necessary ingredient to extend this classification to compact Kaehler manifolds is the use of suitable small deformations of the complex structure. This is joint work with M. Manjarin and M. Nicolau.