Systems based on mechanical resonators are among the most sensitive force probes in existence. The innate ability of mechanical devices to interact with almost any other system through a variety of interaction forces is, at once, a blessing and a curse. Tailoring these interactions is an act of balance and is at the very heart of the research field of cavity optomechanics and, as a result, of the work presented here. In this thesis, I report on the development of mechanical resonators with exceedingly low internal dissipation, and their application in cavity optomechanics. These devices are based on highly tensioned silicon nitride membranes, whereupon we pattern a phononic crystal structure. Introducing a geometric defect in the central region of this crystal structure allows for spatial localisation of vibrations within the frequency range of the phononic bandgap. These vibrational modes exhibit a distinct departure from regular clamping conditions of a square membrane resonator. Specifically, these resonators, which are attached to silicon substates, do not experience the rigid clamp otherwise present at the interface between a membrane and the underlying substate. Instead, the evanescent decay of the localised vibration modes into the phononic crystal patterned membrane structure constitutes an effective "soft" clamping condition. Using this technique we realise soft clamped membrane resonators with mechanical quality factors in excess of 1B at moderate cryogenic temperatures. With the highest reported quality factor of 1.555B at 1.28 MHz frequency, these devices are well suited for experiments in quantum optomechanics. Upon embedding such a device inside a high-finesse optical cavity, we coupling the transverse motion of a localised vibrational mode to the cavity field. Due the weak coupling of these mechanical devices to their phonon thermal bath, we observe a strong influence of the radiation pressure back-action force on the resonator motion. Particularly, we demonstrate strong ponderomotive squeezing of light below the vacuum noise level, with (-2.89 +/- 0.31) dB as the highest degree of measured squeezing. Furthermore, our results suggest that we can indeed prepare these devices close to their motional ground state. Due to the long coherence time of our mechanical devices, as well as the high detection efficiency, our optomechanical system lends itself for a number of quantum optical protocols, including the generation, storage and tomography of single excitations states.