The following preprint is available, in ps: M. Grandis - R. Pare´ Adjoints for double categories Dip. Mat. Univ. Genova, Preprint 476 (2003), 40 p. http://www.dima.unige.it/~grandis/Dbl.Adj.ps Abstract. After 'Limits in double categories' (Cahiers Topologie Geom. Differentielle Categ. 40 (1999), 162-220), we pursue our study of the general theory of weak double categories, addressing adjunctions and monads. A general 'double adjunction', which appears often in concrete constructions, has a *colax* double functor left adjoint to a *lax* one. This cannot be viewed as an adjunction in some bicategory, because lax and colax morphisms do not compose well and do not form one. However, such adjunctions live within an interesting construct, the strict double category of weak double categories, with horizontal and vertical arrows provided by the lax and colax double functors, respectively, and a suitable notion of double cell. Marco Grandis Dipartimento di Matematica Universita` di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it tel: +39.010.353 6805 fax: +39.010.353 6752 http://www.dima.unige.it/~grandis/