  This is a sequel to [Ca01]=math.AG/0110051. We define the bimeromorphic {\it
category} of geometric orbifolds. These interpolate between (compact K\" ahler)
manifolds and such manifolds with logarithmic structure. These geometric
orbifolds are considered from the point of view of their geometry, and thus
equipped with the usual invariants of varieties: morphisms and bimeromorphic
maps, differential forms, fundamental groups and universal covers, fields of
definition and rational points. The most elementary properties, directly
adapted from the case of varieties without orbifold structure, are established
here. The arguments of [Ca01] can then be directly adapted to extend the main
structure results to this orbifold category. We hope to come back to deeper
aspects later. The motivation is that the natural frame for the theory of
classification of compact K\" ahler (and complex projective) manifolds includes
at least the category of orbifolds, as shown in [Ca01] by the fonctorial
decomposition of {\it special} manifolds as tower of orbifolds with either
$\kappa_+=-\infty$ or $\kappa=0$, and also, seemingly, by the minimal model
program, in which most proofs work only after the adjunction of a "boundary".
  Also, fibrations enjoy in the bimeromorphic category of geometric orbifolds
extension properties not satisfied in the category of varieties without
orbifold structure, permitting to express invariants of the total space from
those of the generic fibre and of the base. For example, the natural sequence
of fundamental groups is exact there; also the total space is special if so are
the generic fibre and the base. This makes this category suitable to lift
properties from orbifolds having either $\kappa_+=-\infty$ or $\kappa=0$ to
those which are special.
