In the present paper, degeneration phenomena in conformal field theories are
studied. For this purpose, a notion of convergent sequences of CFTs is
introduced. Properties of the resulting limit structure are used to associate
geometric degenerations to degenerating sequences of CFTs, which, as familiar
from large volume limits of non-linear sigma models, can be regarded as
commutative degenerations of the corresponding ``quantum geometries''.
As an application, the large level limit of the A-series of unitary Virasoro
minimal models is investigated in detail. In particular, its geometric
interpretation is determined.