There are basically only two classes of rigorously defined CFTs: the rational theories (RCFT) and the lattice CFTs. Considerations of this paper correspond to the case of sigma models which produce neither of these. The description of sigma models as path integrals corresponding to certain Lagrangians did not give yet a mathematically satisfactory construction. As we will explain below, there is an alternative way to speak about sigma models in terms of degenerations of CFTs.