Broadband dielectric measurements were carried out in the supercooled as well as in the glassy state of two very important disaccharides: trehalose and sucrose. Multiple relaxation processes were observed. Above the glass transition temperatures of examined disaccharides structural relaxation of cooperative origin was detected, where in the glassy state more local motions (secondary modes) appeared. Our data were discussed in light of the findings reported by other groups. We pointed out that sample preparation might impact mobility and, thus, dielectric loss spectra in a significant way. Consequently, it may lead to misinterpretation of the dielectric relaxation processes. Moreover, impact of physical aging and pressure on dynamics of two secondary relaxation processes observed in the glassy state of trehalose and sucrose has been investigated. Additionally, we have demonstrated that, in contrast to the calorimetric measurements (DSC), activation energies of the β- and γ-relaxation processes observed in the glassy state of sucrose and trehalose do not change as a result of physical aging. Finally, we found out that the β-relaxation process slows down as pressure increases. We interpreted this fact in view of increasing rigidity of the structures of disaccharides.