The birational classification of projective manifolds with holomorphic tangent vector fields was developed by F. Severi, R. Hall and D. Lieberman. The Albanese mapping allows one to make this classification biholomorphic in the non-uniruled case, extending Calabi‘s structure theorem for varieties with trivial canonical bundle. These results may be extended to compact Kähler manifolds, using small deformations of the complex structure. They show that the study of the dynamics of holomorphic vector fields in them reduces to the case of rational varieties.