Graphene is popularly described as a 2D crystal of carbon atoms with charge carriers that behave as massless Dirac Fermions in the honeycomb lattice. The exotic nature of these quasi-particles is at the heart of several interesting phenomena, including the unconventional quantum Hall effect and Klein tunnelling. In this talk, a different aspect of graphene will be explored. Graphene also represents a two-dimensional flexible elastic system in a space of higher dimension. The morphology of graphene films is uniquely coupled to its electronic degrees of freedom. This offers a fascinating opportunity to study the interplay of strain, curvature and electronics on the thinnest possible membrane. Our work provides insight into the physics of atomically-thin elastic membranes in contact with surfaces having tuneable spatial structure. The counterintuitive possibility of inducing flatter phases on rougher substrates and its significance for electron transport will be discussed. The local curvatures of graphene membranes also determine the distribution of deliberately introduced atomic-scale defects. In such membranes, carriers are transported by a new mechanism previously unreported in any condensed matter system. Last but not least, we show that the out-of-plane distortion of the planar carbon-bonds allows spins to be manipulated in the absence of magnetic elements or externally applied magnetic fields.