This thesis presents a quantum-state-resolved molecular beam study of the non-reactive scattering of methane (CH4) from a Ni(111) surface. It is one of the first experimental investigations in which the internal quantum state distribution of a polyatomic molecule is measured after surface scattering. The quantum state populations of scattered CH4 were probed by selective rovibrational excitation using a high-power continuous-wave (cw) infrared (IR) laser in combination with a cryogenic bolometer. This technique is introduced as Bolometric detection with Infrared Laser Tagging (BILT) and its experimental realization is described in detail. Example data illustrates the capabilities and the performance of the method. Scattering experiments were conducted in a near-specular geometry at grazing incidence 65°) and exit angles (70°). The surface temperature was in all cases 673 K. Two aspects of the scattering dynamics of CH4 at Ni(111) were investigated. First, the fate of initial vibrational energy in the gas-surface encounter between CH4 and Ni(111) was studied in a state-to-state scattering experiment. Here, incident CH4 was prepared with one quantum of the anti-symmetric C-H stretch vibration (v3) and in rotational state J=1 by coherent IR pumping. The results include the first observation of vibrational energy redistribution in the direct scattering of a molecule from a surface. Specifically, a portion of the CH4 molecules, which were initially in the v3 state, were detected in the symmetric C-H stretch state (v1) after scattering. The probability for this vibrationally inelastic process is about 40% compared to the vibrationally elastic process in which CH4 remains in the initially prepared v3 state. This branching ratio is insensitive to changes in incidence kinetic energy in the range 100-370 meV. Rotational excitation is in all cases significant, where molecules that underwent v3-to-v1 conversion carry away an increased amount of rotational energy. The results are discussed in the context of previously observed mode-specific reactivity in this gas-surface system. Second, the rotational excitation of scattered CH4 in its vibrational ground state was investigated. The scattering is likewise direct and the final rotational state distributions are non-Boltzmann, revealing a propensity for scattering into low-J states. Extended analysis of the rotational-state-resolved angular distributions and the Doppler-broadened absorption profiles suggest that, at low incidence kinetic energies, rotational excitation is dominated by energy transfer from the surface, i.e. phonon annihilation. This conclusion is supported by classical scattering simulations, which recover the rotational excitation at low incidence kinetic energies. However, they strongly overestimate the efficiency of translational-to-rotational energy transfer. The highly detailed scattering data obtained in this work can serve as stringent test of multi-dimensional dynamical models of this prototypical gas-surface reaction, thereby paving the way toward a predictive understanding of heterogeneous catalysis. This work also proves that BILT detection using state-of-the-art IR light sources is sufficiently sensitive to enable state-to-state surface scattering experiments on polyatomic molecules, opening the possibility to study their dynamics at surfaces with unprecedented detail.