This thesis reports research activity on suspended graphene membranes. Scientific results in the form of peer-reviewed publications are presented, along with supporting information to provide context, detailed experimental procedures, and recommendations of future work. The four papers cover a wide variety of topics, but are linked by common experimental sample fabrication techniques. Understanding the mechanical properties of suspended graphene membranes is crucial to the development of graphene nano-electromechanical devices. In the first presented paper, PeakForce QNM (quantitative nanomechanical mapping) atomic force microscopy imaging was used to rapidly map the nanomechanical properties of a range of suspended graphene membranes. The force-displacement behaviour of monolayer graphene extracted from the peak force imaging map was found to be comparable to that taken using standard nanoindentation. By fitting to a simple elastic model, the two-dimensional elastic modulus was measured at around 350Nm-1, corresponding to a Young's modulus of around 1 TPa. The second paper examines the near-IR light-matter interaction for graphene integrated cavity ring resonators based on silicon-on-insulator (SOI) racetrack waveguides. Fitting of the cavity resonances from the predicted transmission spectra reveal the real part of the effective refractive index for graphene, neff = 2.23 ± 0.02 and linear absorption coefficient, alphagTE = 0.11 ±0.01dB micro metre-1. The evanescent nature of the guided mode coupling to graphene at resonance depends strongly on the height of the graphene above the cavity, which places limits on the cavity length for optical sensing applications. Twisted-bilayer graphene (tBLG) exhibits van Hove singularities in the density of states that can be tuned by changing the twisting angle θ. In the third paper, θ-defined tBLG was produced and characterized using optical reflectivity and resonance Raman scattering. This represents the first reported fabrication of a rationally designed (twist engineered) tBLG structure. The θ-engineered optical response is shown to be consistent with persistent saddlepoint excitons. Separate resonances with Stokes and anti-Stokes Raman scattering components can be achieved due to the sharpness of the two-dimensional saddle-point excitons, similar to what has been previously observed for one-dimensional carbon nanotubes. The excitation power dependence for the Stokes and anti-Stokes emissions indicate that the two processes are correlated and that they share the same phonon. Nano-patterned and suspended graphene membranes find applications in electronic devices, filtration and nano-pore DNA sequencing. However, the fabrication of suspended graphene structures with nanoscale features is challenging. In the fourth and final paper, the direct patterning of suspended membranes consisting of a graphene layer on top of a thin layer of hexagonal boron nitride which acts as a mechanical support is demonstrated for the first time, using a highly focused electron beam to fabricate structures with extremely high resolution within the scanning transmission electron microscope. The boron nitride support enables the fabrication of stable graphene geometries which would otherwise be unachievable, by preventing intrinsic strain in graphene membranes from distorting the patterned features after areas are mechanically separated. Line cuts with widths below 2 nm are reported. It is also demonstrated that the cutting can be monitored in-situ utilising electron energy loss spectroscopy (EELS).