  We establish that the relevant geometric data for the target space
description of world sheet topological defects are submanifolds - which we call
bi-branes - in the product M1 x M2 of the two target spaces involved. Very much
like branes, they are equipped with a vector bundle, which in backgrounds with
non-trivial B-field is actually a twisted vector bundle. We explain how to
define Wess-Zumino terms in the presence of bi-branes and discuss the fusion of
bi-branes.
  In the case of WZW theories, symmetry preserving bi-branes are shown to be
biconjugacy classes. The algebra of functions on a biconjugacy class is shown
to be related, in the limit of large level, to the partition function for
defect fields. We finally indicate how the Verlinde algebra arises in the
fusion of WZW bi-branes.
