Hydrates of gas are non-stoichiometric inclusion compounds constituted of water and gas. Therein, the water molecules are hydrogen-bonded and form three-dimensional crystalline networks incorporating different kinds of polar or nonpolar guest gas molecules. Those networks are clathrate structures at relatively low pressures and non-clathrate, filled ice structures at very high pressures (in the GPa range and above). Gas hydrates spontaneously form whenever water and a hydrate-forming gas are in contact at high pressure and/or low temperature. In those systems the guest molecules may perform many different dynamical processes: rotation, diffusive or quantized confined motion, cage-to-cage hopping, and translational diffusion at the structure interface. The guest dynamics is the key for stabilizing those structures and therefore to understand the process of clathrates formation as well as gas exchange processes within the structures. Investigating the guest dynamics is thus a very interesting topic from a fundamental point of view (e.g. to understand water-gas interaction) and highly relevant to the technological issues involving gas hydrates (e.g. energy recovery, flow assurance, gas transportation and storage). This thesis focuses on the dynamics of the guest molecules in the hydrates of methane and hydrogen under high pressure, over a wide range up to 150 GPa. Pressure is a key parameter in the study of gas hydrates as it induces substantial variations in the water-gas distances as well as complete structural rearrangements. Furthermore, gas hydrates could be major constituents of the interiors of icy bodies of the Universe and therefore their high-pressure properties are of interest to planetary modeling. We use inelastic and quasielastic neutron scattering, and Raman spectroscopy measurements on laboratory-produced methane hydrate and hydrogen hydrate samples. Interpretation of the Raman data is supported by molecular dynamics simulations. Complementary neutron and synchrotron x-ray diffraction measurements are used to monitor the system structure and structural changes. Different types of high-pressure cells are employed to span such a wide pressure range with different experimental techniques, namely a gas pressure cell, a Paris-Edinburgh cell, and a diamond anvil cell. Three main topics are treated. In the first part, we measure the classical translational diffusion of methane molecules at the interface of two clathrate structures by quasielastic neutron scattering at 0.8 GPa. We find a remarkably fast diffusion, faster than that expected in pure methane at comparable pressure and temperature. In the second part, we study the vibrational dynamics, orientational ordering, and distortion of methane molecules embedded in methane hydrate at extremely high pressures by simulations up to 45 GPa and Raman spectroscopy up to 150 GPa. We observe complete locking-in of the rotations at about 20 GPa, and no hints of decomposition up to the highest investigated pressure. Finally, we investigate the quantum roto-translational dynamics of hydrogen molecules nanoconfined in two different hydrate structures by inelastic neutron scattering at pressures up to 1.4 GPa and temperatures below 50 K. Among other things, we report the first experimental observation of quantized translational dynamics for H2 and D2 in the large cage of clathrate structure II.