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The growth and exfoliation of two-dimensional (2D) materials have led to thecreation of edges and novel interfacial states at the juncture between crystalswith different composition or phases. These hybrid heterostructures (HSs) canbe built as vertical van der Waals \emph{stacks}, resulting in a 2D interface,or as \emph{stitched} adjacent monolayer crystals, resulting in one-dimensional(1D) interfaces. Although most attention has been focused on vertical HSs,increasing theoretical and experimental interest in 1D interfaces is evident.In-plane interfacial states between different 2D materials inherit propertiesfrom both crystals, giving rise to robust states with unique 1D non-parabolicdispersion and strong spin-orbit effects. With such unique characteristics,these states provide an exciting platform for realizing 1D physics. Here, wereview and discuss advances in 1D heterojunctions, with emphasis on theoreticalapproaches for describing those between semiconducting transition metaldichalcogenides $MX_{2}$ (with $M$=Mo, W and $X$= S, Se, Te), and how theinterfacial states can be characterized and utilized. We also address how theinterfaces depend on edge geometries (such as zigzag and armchair) or strain,as lattice parameters differ across the interface, and how these featuresaffect excitonic/optical response. This review is intended to serve as aresource for promoting theoretical and experimental studies in this rapidlyevolving field.

Lateral heterostructures and one-dimensional interfaces in 2D transition metal dichalcogenides