Atomic scale characterization of mismatched graphene layers

•Review of STM/STS of graphene with various degree of coupling.•Review of vertically twisted graphene with respect with each other.•Review of Landau levels in graphene layers weakly decoupled electronically.•Review of laterally twisted graphene forming grain boundaries.

In the bourgeoning field of two dimensional layered materials and their atomically thin counterparts, it has been established that the electronic coupling between the layers of the material plays a key role in determining its properties [1,2]. We are just beginning to understand how each material is unique in that respect while working our way up to building new materials with functionalities enabled by interlayer interactions. In this review, we will focus on a system that despite its apparent simplicity possesses a wealth of intriguing physics: layers of graphene with various degree of coupling. The situations discussed here are graphene layers vertically twisted with respect with each other, weakly decoupled electronically and laterally twisted forming grain boundaries. We emphasize experiments that atomically resolve the electronic properties.