  Hermitian bundle gerbes with connection are geometric objects for which a
notion of surface holonomy can be defined for closed oriented surfaces. We
systematically introduce bundle gerbes by closing the pre-stack of trivial
bundle gerbes under descent. Inspired by structures arising in a representation
theoretic approach to rational conformal field theories, we introduce geometric
structure that is appropriate to define surface holonomy in more general
situations: Jandl gerbes for unoriented surfaces, D-branes for surfaces with
boundaries, and bi-branes for surfaces with defect lines.
