This is to announce two preprints, which take on our study of weak double categories and deal with Kan extensions. The first two parts, dealing with limits and adjoints, respectively, where published in "Cahiers" (1999, 2004) and can also be found on my server. M. Grandis ________________ M. Grandis - R. Pare, Kan extensions in double categories (On weak double categories, Part III), Dip. Mat. Univ. Genova, Preprint 553 (2007). http://www.dima.unige.it/~grandis/Dbl3.pdf (ps) ________________ ---, Lax Kan extensions for double categories (On weak double categories, Part IV), Dip. Mat. Univ. Genova, Preprint 554 (2007). http://www.dima.unige.it/~grandis/Dbl4.pdf (ps) ________________ Abstracts. Part III. This paper deals with Kan extensions in a weak double category. Absolute Kan extensions are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category. Part IV. Right Kan extensions for weak double categories extend double limits and other constructions, called vertical companions and vertical adjoints, studied in previous papers. We prove that these particular cases are sufficient to construct all pointwise unitary lax right Kan extensions, along those lax double functors which satisfy a Conduche type condition. Double categories 'based on profunctors' are complete, i.e. have all such constructions, while the double category of commutative squares on a complete category is not, in general.