  In this paper we combine the principled approach to modalities from
multimodal type theory (MTT) with the computationally well-behaved realization
of identity types from cubical type theory (CTT). The result -- cubical modal
type theory (Cubical MTT) -- has the desirable features of both systems. In
fact, the whole is more than the sum of its parts: Cubical MTT validates
desirable extensionality principles for modalities that MTT only supported
through ad hoc means. We investigate the semantics of Cubical MTT and provide
an axiomatic approach to producing models of Cubical MTT based on the internal
language of topoi and use it to construct presheaf models. Finally, we
demonstrate the practicality and utility of this axiomatic approach to models
by constructing a model of (cubical) guarded recursion in a cubical version of
the topos of trees. We then use this model to justify an axiomatization of
L\"ob induction and thereby use Cubical MTT to smoothly reason about guarded
recursion.
