Heterostructures unconstrained by epitaxy have generated considerable excitement due to the discovery of emergent properties-properties not found in either constituent. Heterostructures enable the surfaces on either side of two-dimensional (2D) layers to be used to systematically investigate phenomena such as superconductivity and magnetism in the 2D limit. The ability to choose constituents facilitates the prediction of emergent properties created by the unusual coordination environments at incommensurate interfaces. There have already been many reviews on heterostructures, focusing on a variety of topics that reflect the diverse interest in this area as well as the potential for new technologies. Hence this review focuses mainly on the synthesis and structural characterization of heterostructures containing transition metal dichalcogenides (TMD). This review only briefly discusses 2D materials and TMD/TMD heterostructure devices and the performances that have been achieved. This review provides a historical context for the rapid development of this field and discusses proposed mechanisms for emergent properties. Up to now, the materials used in heterostructures have mainly been materials with 2D structures, as these compounds can be easily cleaved into ultrathin layers. This review discusses the expansion of heterostructure constituents to include materials that do not have 2D structures. Structural changes and charge redistribution between adjacent (or even more distant) layers are likely to be larger for 3D constituents than with 2D constituents based on known misfit layer compounds. Systematic changes in properties with layer thickness, layer sequences, and the identity of constituents will increase our understanding of emergent properties and how they can be optimized.