Epitaxy is the growth of crystalline material on top of a substrate with a matching (or very close to it) crystal structure. For example, it is possible to grow InAs epitaxially on top of GaSb, or SiGe epitaxially on top of Si. The idea is that the lattice of the underlying material guides the growth of the new layers of atoms, and if the lattice mismatch isn't too bad and the conditions are right, you can get extremely high quality growth (that is, with nearly perfect structure). The ability to grow semiconductor films epitaxially has given us a ton of electronic devices that are everywhere around us, including light emitting diodes, diode lasers, photodiodes, high mobility transistors, etc. Note that when you grow, say, AlGaAs epitaxially on a GaAs substrate, you end up with one big crystal, all covalently bonded. You can't readily split off just the newly grown material mechanically. If you did homoepitaxy, growing GaAs on GaAs, you likely would not even be able to figure out where the substrate ended and the overgrown film began.
In this paper (sorry about the Nature paywall - I couldn't find another source), a group from MIT has done something very interesting. They have shown that a monolayer of graphene on top of a substrate does not screw up overgrowth of material that is epitaxially registered with the underlying substrate. That is, if you have an atomically flat, clean GaAs substrate ("epiready"), and cover it with a single atomic layer of graphene, you can grow new GaAs on top of the graphene (!), and despite the intervening carbon atoms (with their own hexagonal lattice in the way), the overgrown GaAs will have registry (crystallographic alignment and orientation) with the underlying substrate. Somehow the short-ranged potentials that guide the overgrowth are able to penetrate through the graphene. Moreover, after you've done the overgrowth, you can actually peel off the epitaxial film (!!), since it's only weakly van der Waals bound to the graphene. They demonstrate this with a variety of overgrown materials, including a III-V semiconductor stack that functions as a LED.
I found this pretty amazing. It suggests that there may be real opportunities for using layered van der Waals materials to grow new and unusual systems, perhaps helping with epitaxy even when lattice mismatch would otherwise be a problem. I suspect the physics at work here (chemical interactions from the substrate "passing through" overlying graphene) is closely related to this work from several years ago.