We present results of extensive theoretical studies of functionalized carbon nanotubes (CNTs) and graphene layers (GLs). Our studies are based on the ab initio calculations in the framework of the density functional theory (DFT) and provide valuable quantitative predictions that are of importance for design of novel composite materials and functional devices. We have performed calculations fort metallic (9,0) and semiconductor (10,0) and (11,0) single-wall CNTs and epitaxial GLs, functionalized with simple organic molecules, such as -OH, -COOH, -NH, -NH2 and -CH3. We have determined the changes in the geometry, adsorption and binding energies, the Young’s modulus, and the band structure as a function of the density of the adsorbed molecules. The theoretical elastic moduli of CNTs and GLs agree very well with experimental values. We observe characteristic effects such as rehybridization of the bonds induced by fragments attached to graphene and nanotubes and deformation of systems that results further in decrease of the Young’s modulus. However, the functionalized carbon systems remain strong enough to be used as reinforcement in composite materials. We have also determined the critical density of molecules that could be adsorbed on the surface of CNTs and GLs. We show that the functionalization of the single graphene layer can open its electronic gap, which could be utilized in graphene devices. Our calculations reveal that the –NH radicals exhibit the strongest cohesion to GLs and CNTs. Further, we determine the critical density of the -NH fragments that leads to the closing of the band gap in functionalized CNT. We also show how to engineer the magnitude of the band gap by functionalizing graphene with -NH2, -NH, -COOH and –CH3 groups of various concentrations.