Many experimental and theoretical studies of adsorption processes on semiconductors have employed Si(001) as a prototype substrate surface. The ionic group IV compound semiconductor silicon carbide (SiC), which is of large fundamental interest and of high application potential for electronic devices and sensors, offers particularly interesting new degrees of freedom for adsorption because of its different lattice constant and its rich variety of surface reconstructions giving rise to very amazing adsorption behaviour. For example, molecular hydrogen adsorbs dissociatively on SiC(001)-c(4x2) while it does not adsorb on SiC(001)-(3x2) at room temperature although both surfaces are characterized by similar surface dimers. Likewise the covalent attachment of organic layers to the SiC(001)-(3x2) surface, contrary to the Si(001)-(2x1) surface, gives rise to structurally well-defined configurations that can play a key role in organic functionalization. Incorporating new functionalities at SiC surfaces by adsorption of hydrogen or hydrocarbons requires a detailed understanding of the adsorbate-surface structure and its properties. In this talk adsorption of hydrogen, acetylene, ethylene and benzene on SiC(001) surfaces will be discussed. Scrutinizing a number of conceivable reaction scenarios within first-principles theory allows us to follow surface reactions on a microscopic level giving access to reaction mechanisms, transition states, barrier heights and adsorption energies and thus to identify the physical origin and nature of the peculiar adsorption behavior of SiC(001) surfaces. The theoretical results will be reviewed and discussed in comparison with available experimental data, as well as results for related adsorption processes on the Si(001)-(2x1) surface.