  The aim of this paper is to review and discuss in detail local aspects of
principal bundles with groupoid structure. Many results, in particular from the
second and third section, are already known to some extents, but, due to the
lack of a ``unified'' point of view on the subject, I decided nonetheless to
(re)define all the main concepts and write all proofs; however, some results
are reformulated in a more elegant way, using the division map and the
generalized conjugation of a Lie groupoid. In the same framework, I discuss
later generalized groupoids and Morita equivalences from a local point of view;
in particular, I found a (so far as I know) new characterization of generalized
morphisms coming from nonabelian ech cohomology, which allows one to view
generalized morphisms as a generalization of classical descent data. I found
also a factorization formula for the division map, which is the crucial point
in the local formulation of Morita equivalences.
