There are several examples of covalently linked 2DPs which include the individual layers or sheets of graphite (called graphenes), MoS2, (BN)x and layered covalent organic frameworks. As required by the above definition, these sheets have a periodic internal structure.
A well-known example of a 2D polymer is graphene; whose optical, electronic and mechanical properties have been studied in depth.  Graphene has a honeycomb lattice of carbon atoms that exhibit semiconducting properties.  A potential repeat unit of graphene is a sp2-hybridized carbon atom. Individual sheets can in principle be obtained by exfoliation procedures, though in reality this is a non-trivial enterprise.
Molybdenumdisulfide can exist in two-dimensional, single or layered polymers where each Mo(IV) center occupies a trigonal prismatic coordination sphere.
Boron nitride polymers are stable in its crystalline hexagonal form where it has a two-dimensional layered structure similar to graphene. There are covalent bonds formed between boron and nitrogen atoms, yet the layers are held together by weak van der Waals interactions, in which the boron atoms are eclipsed over the nitrogen.
Two dimensional covalent organic frameworks (COFs) are one type of microporous coordination polymer that can be fabricated in the 2D plane. The dimensionality and topology of the 2D COFs result from both the shape of the monomers and the relative and dimensional orientations of their reactive groups. These materials contain desirable properties in fields of materials chemistry including thermal stability, tunable porosity, high specific surface area, and the low density of organic material. By careful selection of organic building units, long range π-orbital overlap parallel to the stacking direction of certain organic frameworks can be achieved.
Many covalent organic frameworks derive their topology from the directionality of the covalent linkages, thus small changes in organic linkers can dramatically affect their mechanical and electronic properties. Even small changes in their structure can induce dramatic changes in stacking behavior of molecular semiconductors.
Porphyrins are an additional class of conjugated, heterocyclic macrocycles. Control of monomer assembly through covalent assembly has also been demonstrated using covalent interactions with porphyrins. Upon thermal activation of porphyrin building blocks, covalent bonds form to create a conductive polymer, a versatile route for bottom-up construction of electronic circuits has been demonstrated.