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In this thesis single- (SLG) and bi-layer (BLG) graphene on hexagonal boron nitride (hBN) are investigated by means of Raman spectroscopy and transport experiments. The hBN-graphene heterostructures are fabricated by a state-of-the-art transfer process of exfoliated graphene on exfoliated hBN. The high quality structures feature low residual charge carrier densities of n∗ ≈ 5×10$^{10}$ cm$^{−2}$ and high carrier mobilities of μ ≈ 70,000 cm$^2$/Vs. Thermal annealing experiments show that these superior sample characteristics can even be maintained after substantial heat treatments. A subsequent experiment investigates the scattering mechanism limiting the carrier mobility in SLG. By correlating Raman spectroscopy to transport measurements we find new evidence that strain variations are the scattering source for charge carriers in SLG. The limits of carrier mobility are further investigated in case of BLG. A thorough analysis of the weak localization effect leads to the conclusion that intravalley scattering originating from nanometer-scale strain variations is the most likely source for the mobility limiting scattering events. To obtain direct evidence for this hypothesis we further investigate the relation of n∗ to μ for a large number of samplesand observe the predicted dependence for strain variations as the main scattering source. In consequence, we conclude that strain fluctuations are the most probable source for carrier scattering in both SLG and BLG.In a following set of experiments the opening of the band gap in BLG in dependence of the applied perpendicular electric field is investigated. The investigation particularly focuses on the origin of the sub gap conductance which leads to a maximum resistance on the order of ten megaohm. Temperature dependent transport and bias spectroscopy measurements show that this non-vanishing conductance is mediated by variable range hopping through a low number of hopping states. By developing a simple model we can unambiguously rule out long range potential disorder as the origin of the hopping states. Further measurements of a ”corbino-like” device suggest the presence of AB/BA stacking domain walls in the bulk of the investigated BLG. A final experiment investigates the influence of the rough edges in graphene nanostructures i.e. quantum dots (QDs). The obtained results show that edge roughness is the dominant source of disorder in QDs on hBN with diameters below 100 nm.