  We present a framework that explains the strong connection in 2D materials
between mechanics and electronic structure, via dislocation theory. Within this
framework, Moir\'e patterns created by layered 2D materials may be understood
as dislocation arrays, and vice versa. The dislocations are of a unique type
that we describe as van der Waals dislocations, for which we present a complete
geometrical description, connected to both stretch and twist Moir\'e patterns.
A simple computational scheme, which reduces the complexity of the electronic
interaction between layers in order to make the problem computationally
tractable, is introduced to simulate these dislocation arrays, allowing us to
predict and explain all of the observed Moir\'e patterns in 2D material systems
within a unique framework. We extend this analysis as well to defects in
Moir\'e patterns, which have been reported recently, and which are the result
of defects of the same symmetry in the constituent 2D material layers. Finally,
we show that linear defects in the Moir\'e space can be viewed as
unidimensional topological states, and can be engineered using our framework
