The deformation theory of categories with structure (e.g. braided monoidal categories) is bound to geometric topology by two ties. On the one hand, categorification, first proposed on the basis of consideration of algebraic constructions for TQFT's and related QFT's. In the first instance, categorification, like quantization, is a relationship between two different mathematical structures, which only in special cases can be made into a construction. In the second instance, categorification becomes the problem of classifying structures for which a specific instance of this relationship holds--which problem is naturally deformation theoretic in nature. On the other hand, the well-understood instances of categorical deformation theory, for monoidal categories, functors and natural transformations, give rise to a deformation theory for ribbon categories, which is, in fact the tangle-theoretic version of (framed) Vassiliev theory.