Two-dimensional atomic sheets of graphene represent a new class of nanoscale materials with potential applications in electronics. However, exploiting the intrinsic characteristics of graphene devices has been problematic due to impurities and disorder in the surrounding dielectric and graphene/dielectric interfaces. Recent advancements in fabricating graphene heterostructures by alternately layering graphene with crystalline hexagonal boron nitride (hBN), its insulating isomorph, have led to an order of magnitude improvement in graphene device quality. Here, recent developments in graphene devices utilizing boron–nitride dielectrics are reviewed. Field-effect transistor (FET) characteristics of these systems at high bias are examined. Additionally, existing challenges in material synthesis and fabrication and the potential of graphene/BN heterostructures for novel electronic applications are discussed.