  In this paper, we give a general axiomatization of anomalies in closed and
open conformal field theories. In particular, we generalize Segal's notion of
modular functor to a setting where the ``set of labels'' is a 2-vector space.
In the case of open conformal field theory, the ``set of $D$-branes'' is a
3-vector space. We also define a ``topological group completion'' of the
symmetric bimonoidal category of finite-dimensional vector spaces, and propose
it as a candidate for labelling conformal field theories whose modular functors
are super-vector spaces.
