Nature has always been inspiration and resource for new technical developments that otherwise would not have been possible. Taking a close look at almost every aspect of daily life and technology it is obvious that, even nowadays, most of the materials are based on natural occurring raw materials and concepts. The rise of quantum mechanics, which is the theory of objects compared to the size of atoms, allowed an in-depth understanding of the physical properties of materials, and also paved the way for optimized or even tailored properties of specific technological applications. The naturally occurring graphite is a highly anisotropic allotrope of carbon. It is built from stratified atomically thin layers of carbon atoms, which are strongly bonded in a honeycomb lattice within each layer. In contrast, the bonding between the layers is only weak due to van der Waals interactions. The theoretical study of the electronic properties of graphite was initiated by Wallace back in the 1940s. Surprisingly, the calculations also revealed remarkable electronic properties of single layer graphite, which was later called graphene. In particular, it was shown that due to the hexagonal symmetry of the carbon atoms within the layer the valence electrons resemble the linear dispersion of massless particles, which led to a strong interest in this material. The first known extraction of a graphene layer has been achieved in 1962 by Boehm, but the reliable production is possible only since 2004 due to Geim and coworkers. For this achievement, Andre Geim and Konstantin Novoselov have been granted the Nobel prize of physics in 2010. Indeed, very recently the large-scale production of graphene by shear-exfoliation has begun. Since graphene is broadly available, manifold developments and predictions have been made for this new "wonder material". It is well known that the massless valence electrons propagate with a finite (Fermi-) velocity of about 300 times smaller than the speed of light. The electronic bandgap between electrons and holes as well as the density of states vanishes at the Fermi level, which makes graphene a zero-gap semiconductor. Consequently, graphene can be considered as an analog to ultrafast spin-1/2-particles observed in a solid-state system. Since the speed of the electrons is comparably low, predictions of high-energy physics can be studied in a material that is now easily accessible. Among them are, e.g., minimal conductivity, Zitterbewegung, universal optical absorbance, and perfect transmission of carriers at normal incidence through a potential barrier, called the Klein paradox. For graphene a very high electron mobility of about 200 times higher than in bulk-diamond structure silicon has been predicted and consequently, its usefulness as a very fast switch in a transistor has been demonstrated, which is about three orders of magnitude higher than currently available switches based on bulk silicon. Recently, the research on graphene has been heavily supported by one of the flagships of European research, funded with one billion euro for the next ten years, which shows the huge interest in this new material. By all advantages of graphene mentioned so far, there are of course also fundamental problems. Even if graphene can be produced on the large scale, all common technology today is based on silicon. The integration of carbon-based electronics into silicon-based circuits is challenging due to deviations of the lattice constants and electronegativities. For that reason silicene, the silicon-based analog of graphene, which does not exist in nature, has been suggested theoretically and experimentally realized as adsorbate layer. It has been shown that the structural and electronic properties of silicene are very similar to that of graphene, in particular the appearance of massless electrons. The first experimental realization of silicene or at least a monolayer of silicon with hexagonal symmetry on Ag(111) has been achieved by Vogt and coworkers in 2012. However, the strong interaction with the silver surface results in structural distortions of the honeycomb lattice and consequently, the disappearance of massless Dirac fermions in the silicon layer. The fabrication of freestanding silicene or at least silicene with weak substrate interactions is still an open quest and subject of intense research. The experimental realization of graphene and silicene has initiated work on the entire field of two-dimensional (2D) crystals like the elemental crystals germanene and stanene (also called tinene) as well as many compounds like boron nitride (BN), molybdenum disulfide (MoS_2) or SiGe alloys. In particular stanene, the tin-based analog to graphene and silicene, has attracted attention due to the predicted appearance of a new state of matter at its one-dimensional edges, called topological edge states. These edge states are relevant for the realization of the quantum spin Hall effect. In this work we mainly focus on the elemental 2D group-IV honeycomb crystals graphene, silicene, germanene and stanene (or tinene). Starting from a general treatment based on parameter-free first-principles methods in the framework of density functional theory (DFT) the structural properties for all freestanding group-IV honeycomb crystals are determined on equal footing. Under ambient conditions, silicon, germanium and tin tend to form sp3 bonds in crystals instead of planar sp2 bonds as carbon atoms in graphite. Hence, we study the formation of covalent bonds in the associated 2D honeycomb crystals and their impact on lattice constants and bond angles, which give rise to buckled structures with a broken sp2 hybridization. The electronic ground-state properties are investigated by means of the electronic band structure and compared to available experiments on graphene and silicene. A reliable description of the electronic band structure in agreement with experiments also requires an improved description of the complex many-body electron-electron interaction. Hence, quasiparticle effects are approximately taken into account by means of the nonlocal exchange-correlation hybrid functional HSE06. In particular for the heavier elements Si, Ge and Sn the increasing impact of spin-orbit coupling (SOC) on the electronic band structure is incorporated into the theory. Without SOC all group-IV honeycomb crystals are rendered as zero-gap semiconductors like graphene, however, SOC opens a bandgap turning these crystals into insulators. According to Fu and Kane all insulators in two and three dimensions can be classified into trivial and topological ones by means of a Z2 topological index. The 2D group-IV honeycomb crystals silicene, germanene and stanene are also examined with regard to their topological character and identified as topological insulators within the DFT. We investigate the topological phase transition to a trivial insulator due to an applied external electric field perpendicular to the sheet plane as predicted by simpler model calculations. Since theory predicts topologically nontrivial edge states in topological insulators, we further investigate exemplarily the edge states of germanene nanoribbons. Particularly interesting is also the optical absorption of the two-dimensional group-IV honeycomb crystals. Experiments by the group of Geim revealed a comparable high absorbance of A=2.3% of graphene in a broad frequency range up to infrared light. Based on a model, this value has also been determined theoretically and revealed a surprising connection to the Sommerfeld fine-structure constant alpha via the formula A=pi*alpha. The main focus of this work is the accurate description of the frequency-dependent reflection, transmission and absorption of the 2D group-IV honeycomb crystals in order to explain and extend the observations of the universal infrared absorbance in graphene over the entire optical spectra. For that reason, the complex optical conductivity of the sheet crystal over the entire frequency axis is computed beyond the common Dirac cone approximation. As a benchmark the infrared absorption of graphene is determined numerically with remarkable accuracy and compared to experiments and analytical calculations. Besides the optical infrared absorption, the numerical treatment also allows an accurate description of the frequency dependent reflection, transmission and absorption of such 2D crystals at arbitrary angles of incidence and polarization of the incident light. We address the question how atomically thin 2D crystals are incorporated in the theory of classical electrodynamics. Thereby, we also explore in detail the limit beyond the description as infinitely thin 2D sheet crystals commonly applied in literature and support our findings with numerical and analytical calculations. The problem of the 2D group-IV honeycomb crystals is their preparation, which is usually achieved by deposition on a substrate. While the majority of results are presented for freestanding 2D sheet crystals we also address the possible impact of metallic and insulating substrates. Therefore, we provide detailed insights in the interaction between the substrate and the overlayer, for which we consider silicene due to recent experimental progress. Based on these findings, we also investigate several new substrates for the growth of silicene with the goal, that even in the presence of a substrate the 2D sheet crystal may be treated as basically freestanding. The calculations are based on total-energy DFT calculations including van der Waals interactions.