The key idea of Multimodal Categorial Grammars, introduced in the 1990s, is simple: instead of a single family of residuated type-forming operations, one considers multiple families, co-existing in one logic. To discriminate between these families, type-forming operations are decorated with a mode index /i, ⊗i, \i; likewise for the interpreting Merge relations Ri. Each family has the same logical rules (the residuation laws). But they can differ in their structural properties. In particular, they can interact in terms of structural rules that mix different modes.  At the level of the interpretation, such interaction principles introduce frame constraints, similar to the constraints that went with the move from NL to L and LP.