Recent advances in synthetic chemistry made available a new class of fascinating compounds in which a molecule of hydrogen is encased inside the hollow cavity of the highly symmetric fullerene C60. There is a great interest in studying endohedral fullerene complexes owing to the rigidity, symmetry, homogeneity and isolation provided by the carbon cages. On one side these properties result in very detailed and wellresolved spectra even in the condensed phase. On the other side such supermolecules are a unique real-world example of a quantum rotor confined in an almost spherical trap, whose dynamics can be directly and accurately treated from first principles. The study of the quantum dynamics and the analysis of the spectroscopic observations of H2@C60 provide a stepping stone for the characterization of the carbon-hydrogen interaction in curved nanocarbons. In this work symmetry is employed to simplify the analytical treatment of the dynamics of the endohedral hydrogen, by expanding the non-bonding confining potential in terms of spherical multipoles and by using a spherical basis to represent the effective rotational-translational Hamiltonian. The thesis features the first study of an endohedral hydrogen fullerenes, H2@C60, by infrared spectroscopy . The quantization of the translational motion, the high rotational freedom and rotational-translational couplings clearly show up in the infrared spectra. The successful assignment of the IR peaks to transitions between the quantum states of the confined hydrogen by means of the presented theoretical model has lead to the first accurate determination of the molecular vibro-rotational parameters and effective hydrogen-cage potentials in H2@C60. The nuclear spin dynamics is very sensitive to the local environment around the hydrogen molecules. Nuclear magnetic resonance spectroscopy at cryogenic temperatures shows that the symmetry at the center of the cages is reduced by solid state effects and/or occluded impurities. The observations of small but not yet completely understood discrepancies in the lineshapes and relaxations of two samples of H2@C60 stress the influence of the preparation, storage and manipulation on the nuclear spin dynamics of endohedral hydrogen fullerenes.