On a long path of finding appropriate materials to store hydrogen, graphene and carbon nanotubes have drawn a lot of attention as potential storage materials. Their advantages lie at hand since those materials provide a large surface area (which can be used for physisorption), are cheap compared to metal hydrides, are abundant nearly everywhere, and most importantly, can increase safety to existing storage solutions. Therefore, a great variety of theoretical studies were employed to study those materials. After a benchmark study of different van-der-Waals corrections to Generalized Gradient Approximation (GGA), the present Density Functional Theory (DFT) study employs Tkatchenko-Schäffler (TS) correction to study the influence of vacancy and Stone- Wales defects in graphene on the physisorption of the hydrogen molecule. Finally, the impact of different carbon nanotube diameters and geometries (zigzag & armchair confguration) on physisorption energetics and behavior is presented.