This thesis explores numerically structures that can act as phononic orphotonic materials. A key feature of photonic and phononic materials is the existenceof frequency gaps in propagation of electromagnetic waves and elastic wavesrespectively. Initially the functionality of two structures as phononic materials isnumerically examined. Those structures have already been used as photonic materials.The first structure is the well-known layer-by-layer structure and the second is anacoustic strip waveguide onto which is considered one phononic crystal. Fornumerical calculations the Finite Difference Time Domain method was used. Thetransmission spectra and the band structure were calculated. Several differentmaterials such as silicon, epoxy and tungsten were included in this study. It was alsoinvestigated the effect of all the geometric parameters of the structures. The resultsshowed that these structures appear to have very promising features as phononiccrystals. Under certain conditions it may even exists a full three-dimensionalphononic band gap. Considering that those structures are already known for their useas photonic crystals, the belief for their use as both photonic crystals and phononiccrystals becomes valid.Then, again using the Finite Difference Time Domain method, potentialapplications that these structures could have were also examined. Initially it wasinvestigated the potential use of phononic crystals as sensors. The sensitivities ofthese structures were calculated from the changes in the boundaries of the respectivephononic band gaps when a thin film of water (in the case of the humidity sensor) wasadded to the structure or when those structures immersed in a liquid (liquid sensors).Also studied for the first time the three-dimensional layer-by-layer structure forspecific elastodynamic behavior. The results show a high value of the ratio of thelongitudinal to the transverse speed of sound and an ideal pentamode behavior for aspecific frequency range. The results clearly show that the layer-by-layer structurecould be a very important elastodynamic metamaterial.In the next section of this thesis, the phonon density of states of graphene-likematerials such as silicene and germanene is examined using density functional theory.Cases were silicon or germanium atoms on graphene-like structures are replaced byother group IV atoms and how these new structures could perform as nanoscalephononic crystals, creating phononic band gaps in their phonon density of states, arenumerically investigated. Nanotubes were also examined and their similarities,especially for cases with diameters above , with the graphene-like materials werefound.In the final section of this thesis structures which could confine light innanometer areas were numerically examined. A system consisting of two silicon diskswith in plane separation of a few tens of nanometers has been studied first. Thenormalized unitless effective mode volume, Veff, has been calculated for the twolowest whispering gallery modes resonances. The effective mode volume is reduced significantly as the gap between the disks decreases. It is also numerically examined astructure consisting of a circular slot waveguide which is formed into a silicon diskresonator. It is shown that the proposed structure could have high Q resonances thusraising the belief that it is a very promising candidate for optical interconnectsapplications.