  Building on discoveries in graphene and two-dimensional (2D) transition metal
dichalcogenides, van der Waals (VdW) layered heterostructures - stacks of such
2D materials - are being extensively explored with resulting new discoveries of
novel electronic and magnetic properties in the ultrathin limit. Here we review
a class of naturally occurring heterostructures - so called misfits - that
combine disparate VdW layers with complex stacking. Exhibiting remarkable
structural complexity and diversity of phenomena, misfits provide a platform on
which to systematically explore the energetics and local bonding constraints of
heterostructures and how they can be used to engineer novel quantum fabrics,
electronic responsiveness, and magnetic phenomena. Like traditional classes of
layered materials, they are often exfoliatable and thus also incorporatable as
units in manually or robotically stacked heterostructures. Here we review the
known classes of misfit structures, the tools for their single crystal and thin
film synthesis, the physical properties they exhibit, and computational and
characterization tools that are unraveling their complexity. Directions for
future research are also discussed.
