Single layer Mo and Si thin films deposited for different deposition times on c-Si substrates by Ion beam sputtering technique have been characterized by Grazing Incidence X-ray Reflectivity (GIXRR) measurements. Subsequently, few Mo/Si multilayers having 5 bi-layers with different Mo to bi-layer thickness ratio (Gamma) at a fixed bi-layer thickness have been deposited and characterized by GIXRR measurements. Investigation has been carried out on the different behaviors of the interfaces and also variation of the interface width with Gamma has been studied. Finally, 25 bi-layer Mo/Si multilayer samples having the lowest Gamma value and with and without carbon diffusion barrier layer at both the Mo/Si interfaces are deposited and the interfaces have been characterized by hard X-ray and soft X-ray reflectivity measurements. The study shows improvement in reflectivities of the multilayer with carbon diffusion barrier layer from 38% to 54% at 130 angstrom wavelength. To have further insight into the modification of the interface due to introduction of the carbon barrier layer, depth selective study of both the multilayers have been carried out by Grazing Incidence Extended X-ray Absorption Fine Structure measurements using synchrotron radiation and thereby investigating the local structures around Mo atoms in Si bulk region and Si-on-Mo interface. This study confirms the reduction of diffusion of Mo and Si layer into each other when the carbon buffer layers are present at the interfaces, thus reducing the chance of formation of MoSi2.