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This thesis is divided into two parts: the study on graphene-based transistor sand the study on graphene-based atom chips. For the first part, we explore the tunnelling current characteristics of graphene-based field-effect transistors by means of theoretical investigation and computer simulations for various device parameters such as the misalignment angle between graphene layers, the Fermi energies of the graphene electrodes,the hBN barrier thickness, and the device temperature. We focus more on understanding the tunnelling mechanism and what gives rise to it rather than the form of the tunnelling current characteristics itself. We have found that the misalignment between graphene layers gives rise to a lower tunnelling cur-rent because a larger momentum change is required for electron tunnellings. We have found that the tunnelling current does not vary monotonically with the applied gate voltage, i.e. does not follows Ohm’s law, due to a complicated competition among the availability of electronic states, the density of states, and the conservation of momentum. For a device with perfectly aligned graphene layers, we have found that the increasing density of states dominantly determines the magnitude of the tunnelling current: the higher the density of states, the higher the tunnelling current. We have found that increasing hBN barrier thickness decreases the tunnelling current due to the decay nature of the electron wavefunctions. We have found that a resonant tunnelling occurs when the electrostatic energy difference between graphene layers is approximately equal to the energy associated with the sum of the inverse square of the scattering length scale and the square of the difference in momenta between the two Dirac points of the graphene layers. Lastly, we have found that the tunnelling current increases with increasing temperature due to the non-zero occupation of electrons in a higher density of states above and below the Fermi levels, giving rise to an increase in the like-band tunnellings. For the second part, with the knowledge of the electronic properties of graphene and hBN gathered in the first part, we have found that graphene multilayers yield a weaker Casimir-Polder potential and a longer Johnson noise lifetime than metal slabs, showing the performance advantages of graphene-based atom chips over the present generation of atom chips. The theoretical formalism for calculating the Casimir-Polder potential and the Johnson noise lifetime has been developed and can be used for future work.