Recent advancements in colloidal chemistry demonstrate that two-dimensional single-crystalline sheets of semiconductors forming a honeycomb lattice can be synthesized by oriented attachment of semiconductor nanocrystals [1,2]. Inspired by these results, we have performed atomistic tight-binding calculations of the band structure of CdSe [3,4] and HgTe [5] sheets with honeycomb nano-geometry. We have also considered honeycomb super-lattices of quantum dots that could be made using nano-lithography of HgTe layers. In the case of CdSe sheets [3], we predicted that their conduction band exhibits Dirac cones at two distinct energies and nontrivial flat bands. The lowest Dirac conduction band has s-orbital character and is equivalent to the π bands of graphene but with renormalized couplings. The conduction bands higher in energy have no counterpart in graphene; they combine a Dirac cone and flat bands because of their p-orbital character. We also present very recent results on HgTe [5]. We show theoretically that honeycomb lattices of HgTe can combine the effects of the honeycomb geometry and strong spin-orbit coupling. The conduction bands, experimentally accessible via doping, can be described by a tight-binding lattice model as in graphene, but including multi-orbital degrees of freedom and spin-orbit coupling. This results in very large topological gaps (up to 35 meV) and a flattened band detached from the others. Owing to this flat band and the sizable Coulomb interaction, honeycomb structures of HgTe quantum dots constitute a promising platform for the observation of a fractional Chern insulator or a fractional quantum spin Hall phase. [1] W. H. Evers, B. Goris, S. Bals, M. Casavola, J. de Graaf, R. van Roij, M. Dijkstra, and D. Vanmaekelbergh, Nano Lett. 13, 2317 (2013). [2] M. P. Boneschanscher, W. H. Evers, J. J. Geuchies, T. Altantzis, B. Goris, F. T. Rabouw, S. A. P. van Rossum, H. S. J. van der Zant, L. D. A. Siebbeles, G. Van Tendeloo, I. Swart, J. Hilhorst, A. V. Petukhov, S. Bals, and D. Vanmaekelbergh, Science 344, 1377-1380 (2014). [3] E. Kalesaki, C. Delerue, C. Morais Smith, W. Beugeling, G. Allan, D. Vanmaekelbergh, Phys. Rev. X 4, 011010 (2014). [4] C. Delerue, Phys. Chem. Chem. Phys., 2014, doi: 10.1039/C4CP01878H. [5] Beugeling, W. et al, Nat. Commun. 6:6316 doi: 10.1038/ncomms7316 (2015).